Operator averaged

Vm(hi) (brown-out). The highest anticipated operational average input... 

This facility produces 110 m2/h of enameled steel and operates 4000 h/yr. It uses 0.0042 m3 water/m2 of product in coating operations. Average process water flow is 1.69 m3/h for coating operations and 0.466 m3/h for metal preparation. The primary in-place treatment is chemical coagulation and clarification. Clarification can be either settling or dissolved air flotation. 

If necessary, we can introduce other effective operators (e.g. operators averaged over some groups of pairing states). 

In this section we analyze the dynamics of a spin subject to a classical random field and derive the equation of motion for the spin dynamics (the spin-evolution operator), averaged over the fluctuations. Following the discussion of the case with quantum fluctuations, we first analyze the dynamics in a stationary field B and a random field exactly as in the quantum case one can reduce the analysis of the dissipative corrections to the Berry phase accumulated over a conic loop to the problem with a stationary field by going over to a rotating frame. 

Operation Averaging set Net configurational change Net site interchanges (cc isomer) M(A-B)3 analog... 

It is easy to see that the self-energy operator - average electron-electron interaction -can be considered also as a linear superoperator in the space of the matrices it depends on. Indeed, from the point of view of the 2M-dimensional space of spin-orbitals E[/5] acts as a 2M x 2M matrix, so that E[p] is a 2M x 2M matrix constructed after another 2M x 2M matrix p. On the other hand, it easy to see from the definition of the Coulomb and exchange operators in eq. (1.145) that the result of calculating each of them (and thus of the sum of them) taking a sum of two functions p (x x ) + p2(x x ) and/or a product of this function by a number Ap x x ) as its argument yields respectively a sum of the results of the actions of S and the product in the same number as the result of action of S ... 

Alathematical Operations <> average over indicated region... 

Fuel Requirements,— Locomotive cranes are for the most part steam operated and on account of the intermittent nature of their operation average fuel consumption is the most reliable index of power requirements. With coal of average grade, the average fuel requirements for standard sizes of locomotive cranes are ... 

The total intracavity variances are expressed through the stochastic variables using the relationships between normally-ordered operator averages and stochastic moments with respect to the P-function. Restoring the previous phase structure of intracavity interaction, we obtain that V+ = V- = V and... 

The operator s inability to control perfectly all the variables in an experiment leads to indeterminate error. For example, errors in the concentration of the solution made up in a volumetric flask can result from fluctuations in the room temperature. Changes in humidity in the room can also affect the weights involved in the experiment, for example, the weight of a solute which can adsorb water vapor. Indeterminate or random error affects the precision of the experiment. This type of error is treated by the laws of probability. If one determines the concentration of a solution each day on five successive days, the results are the same only within the precision of the experiment. On the basis of experience, the operator averages the results, and then estimates the standard deviation in order to have an indication of the precision of the experiment. 

The partitioning of the molecular space into well-defined disjoint atomic basins is a necessary requirement for translating the chemist s intuition of expressing molecular properties in terms of atomic and group contributions. In this manner, the expectation value of an observable averaged over all the molecular space is equal to that obtained from the sum of the expectation values of this operator averaged over all the atoms in the molecule (or in the extended system or crystal), and one can write... 

From the cyclic invariance of the trace, the operator can be evaluated at any point along the cyclic imaginary-time path (t). This is depicted schematically in Fig. 2, where the centroid variable, which is off the thermal loop, is also shown. It is nontrivial, therefore, to reformulate the rules for calculating an operator average in order that the centroid density be used as the statistical distribution [34, 45]. To illustrate this principle, the case of an operator average is described here in some detail. As the first step, Eq. (2.43) is rewritten as... 

The operator 4(5) is then represented in 2Af-dimensional Fourier space and the cumulant average over the phase-space path fluctuation variables 5(r) is carried out through second order [5]. After performing the inverse Fourier transformation, the final result for the operator average in the phase-space centroid picture is approximated by the classical-like form [5]... 

As noted previously, the roasters eliminate 95% of the chloride. This increases the daily permissable intake of crude zinc oxide, containing 5% chloride, to 22,000 pounds, or 11 tons, per day. Adding the washing process ahead of the toaster and lowering the chloride content below the design concentration of 1% increases the chloride capacity to 55 tons of crude zinc oxide per day. With the present operation averaging 0.35% chloride, the capacity for crude zinc oxide, based on chloride limitations, has increased to 155 tons per day (56,000 tons per year). 

At this moment there appears as striking the analogy between this classical equation of motion and the previously deduced equation of an operator average (observed value)... 

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