Operator independence, determination

What is the net sample rate and its uncertainty This raises the general question of calculating the uncertainty in the result of some mathematical operations on an uncertain number. If we consider two independently determined numbers and their uncertainties (standard deviations), A + [Pg.572]

Under all circumstances, preference will be given to inherently safe design solutions. Safety systems should be designed to operate independent of any other functions and should operate reliably under all conditions determined by the risk analysis (including start-up, shutdown and maintenance and repair situations). 

The independent-electron approximation was discussed in the previous chapter. The molecular wave functions, ifi, are solutions of the Hartree-Fock equation, where the Fock operator operates on tfi, but the exact form of the operator is determined by the wave-function itself. This kind of problem is solved by an iterative procedure, where convergence is taken to occur at the step in which the wave function and energy do not differ appreciably from the prior step. The effective independent-electron Hamiltonian (the Fock operator) is denoted here simply as H. The wave functions are expressed as linear combinations of atomic functions, x-... 

Dissolved amino acids are commonly divided into two pools that must be analyzed separately DFAA exist as individual monomers in solution, while DCAA are defined operationally as additional amino acids liberated by acid hydrolysis. DCAA are thus presumably present mostly as polypeptides, a supposition supported in at least the high molecular weight (HMW) fraction by N-nuclear magnetic resonance (NMR) spectroscopy data (discussed below). The operational nature of the DFAA versus DCAA definitions means that amino acids liberated from difficult matrixes (e.g., humic substances) also could make up a part of DCAA. Total hydrolysable amino acid (THAA) is another term commonly used to denote both pools together, when the sample is hydrolyzed but DFAA are not independently determined. Because the DFAA pool is typically much smaller than DCAA, THAA values are often assumed to be similar to DCAA. 

The CPHF equations are linear and can be determined by standard matrix operations. independent of the electric field (as is normally the case), the first-order HF properly the ... 

Consensus reality reassures us that there are laws of nature determining the way physical space is presented to our awareness --solid bodies tend to remain that way gravity is a law of motion properties of light affect the way in which we perceive objects in space etc. To us, these laws are objectively self evident, which is to say that in our imaginations we conceive of them operating independently of subjective observation. 

A concentration boundary layer theory clearly is needed to relate C to C, so that membrane properties such as L, a, and P can be correlated with R, at various operating conditions. Slso, since ir in Equations 1 and 5 is an independently determined function of C, a boundary ayer theory could correlate the observed filtrate velocity, J (averaged along the fiber length), with average applied pressure AP. For sufficiently high axial flow velocities, C == C, and a major theoretical barrier to data analysis is removeS. Some early work in reverse osmosis ( ) was done with flat-sheet membranes and large feed stream velocities. 

Section II and Table I show that state-independent effective operators can be obtained with norm-preserving mappings K, or with any of the three kinds of non-norm-preserving mappings K, L), K, L), and K, L). This section first proves that the commutation relations between two arbitrary operators cannot generally be conserved upon transformation to any of these state-independent effective operators. A determination is then made of operators whose commutation relations are preserved by at least some state-independent effective operator definitions, and a few applications are then presented. Particular interest is focused on operators which commute with H, including constants of the motion. 

The bottomhole pressure of injection wells was determined according to the known barometric formula. However, for the gas wells a correction was made for the increase in gas density. The steam density was calculated to be 20-30% higher on account of the moisture content, and gas density was assumed to equal 0.7 of air density. Inasmuch as the pressure changes in the production-observation wells were not significant, the effects that the injection wells exerted mutually among themselves could be ignored. In other words, it was assumed that all injection wells operated independently of one another. Before the pressure field map was constructed, all pressures measured in the production-observation wells were brought up to the moment at which steam injection was discontinued. Visual extrapolation was used for that puipose. 

Bi-directional flux of free cholesterol between cells and lipoproteins occurs, and rate constants characteristic of influx and efflux can be measured [17]. The direction of any net transfer of free cholesterol is determined by the relative free cholesterol/phospholipid molar ratios of the donor and acceptor particles. Cholesterol diffuses down its gradient of chemical potential generally partitioning to the phospholipid-rich particle. Such a surface transfer process can lead to delivery of cholesterol to cells. This mechanism operates independently of any lipoprotein internalization by the receptor-mediated endocytosis. The influence of enzymes such as lecithin-cholesterol acyltransferase and hepatic lipase on the direction of net transfer of free cholesterol between lipoproteins and cells can be understood in terms of their effects of the pool sizes and the rate constants for influx and efflux. 

As aiesult, there is a dhcimua. eitlicr the world is " nun-reaF (in the sense tnat the piopcities of partieles are not determined before measurement) or/and there is instantaneous (i.e., faster than light) eommunieation between particles which operates independently of how far apart they are ( non-locality ). 

As we shall see, all the classification approaches below work by using some knowledge or assumptions about how the features interact in order to reduce the number of parameters than need to be specified. One common approach is to transform the original features into a new feature space in which the new features do in fact operate independently. Another approach is to determine which features interact most, model the dependeneies between these and then assume that the other features do in fact operate independently. In our above example, we may decide for instance that while iq and F3 often interact, operates independently, which means we can reduce the size of the lookup table to six entries. 

The main design feature of Ihe IFF is that it assumes the features operate independently. This ensures that the munber of parameters (weights) to be determined is very low, and hence easy to learn from data or set by hand. The formulation doesn t in general suffer from sparse data problems, as there are nearly always enough examples of each feature present in the data. 

Figure 9. The hydrogen transfer system. Hydrogen diffuses out of the main carrier gas stream through the heated palladium tube and is swept to a second detector by a stream of nitrogen and determined there, while the remaining components are determined at the first detector. In the commercial instrument the two sides of a single thermal conductivity detector operate independently as if they were two separate detectors.

In general, sprinkler systems distribute extinguishing agents to the locations where fires occurs. Sprinkler heads operate independently and determine when they should release the extinguishing agent Placement of sprinkler heads, type of head, appropriate agent, and proper maintenance and testing are essential to the success of a system. 

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