Consistency relation

The multidensity Ornstein-Zernike equation (70) and the self-consistency relation (71) actually describe a nonuniform system. To solve these equations numerically for inhomogeneous fluids one needs only an appropriate generalization of the Lowett-Mou-Buff-Wertheim equation (14). Such a generalization, employing the concept of the partial correlation function has been considered in Refs. 34,35. 

The self-consistency relation between densities (cf. Eq. (75)) takes the form... 

Operators, describing process operations, can be declared at any level of abstraction, but they should maintain consistent relations with each other, since they refer to the same process. For example, the top-level operator, (START UP PLANT), could be refined to the following sequence of operators (see also Fig. 5) (START UP RECOVERY), (START,UP REFINING), (START.UP FEED PREPARATION), and (START. UP. REACTION). Clearly, the preconditions of (START.UP.PLANT) are distributed and represent a subset of the preconditions for all four more detailed operations. Similarly, the postconditions, derived from the startup operation of the four sections, should be... 

For small particles supported on thin films of amorphous or microcrystalline materials it is not easy to determine whether there is any consistent correlation between the particle orientation and the orientation of the adjacent locally ordered region of the substrate. For some samples of Pt and Pd on gamma-alumina, for example, nanodiffraction shows that the support films have regions of local ordering of extent 2 to 5 nm. Patterns from the metal particles often contain spots from the alumina which appear to be consistently related to the metal diffraction spots. 

Eqs. (9),(11) the condition eq. (12) in the random phase approximation (RPA) formalism leads to a self-consistency relation between the symmetry potential and the Landau parameter F [26] ... 

The familiar consistency relation V0 = (ae/4xe of the Morse oscillator is seen to be satisfied. Another manifestation of this relation is xe = l/(/V + 1). 

A number of studies reported that several kinetic models can describe rate data well, when based on correlation coefficients and standard errors of the estimates [25,118,131,132]. Despite this, there often is no consistent relation between the equation which gives the best fit and the physicochemical and min-eralogical properties of the adsorbent(s) being studied. Another problem with some of the kinetic equations is that they are empirical and no meaningful rate parameters can be obtained. 

As discussed above, this discrepancy may be caused by classically forbidden electronic transitions—that is, cases in which a proposed hopping process is rejected due to a lack of nuclear kinetic energy. Figure 11c supports this idea by showing the absolute numbers of successful (thick fine) and rejected (thin line) surface hops. In accordance with the initial decay of the adiabatic population, the number of successful surface hops is largest during the first 20 fs. For larger times, the number of rejected hops exceeds the number of successful surface hops. This behavior clearly coincides with the onset of the deviations between the two classically evaluated curves Nk t) and P t). We therefore conclude that the observed breakdown of the consistency relation (42) is indeed caused by classically forbidden electronic transitions. 

The boiling point, refractive index, and density of the olefin derivative of any paraffin were shown, by use of Table III, to stand in the onier of their olefin type. Table X contains the engine data of the olefin derivatives of 2-methylpentane and 3-methylpentane, recorded in the order of their olefin type. No consistent relations between octane numbers or critical compression ratios are obvious—but the blending octane numbers of these branched olefins, as measured by both the research and Motor methods, do generally stand in the order of their type. Two olefins of type III form exceptions, the exceptions being in one case too high and in the other case too low. 

It would appear from the foregoing that the reactions of recombination and disproportionation of radicals fall into a reasonable relation with each other and with abstraction reactions if in the approach of the radical pair, either head-to-head as in recombination or head-to-tail as in disproportionation, there occurs an appreciable contribution of ionic structures to the attractive potential of the radical pair. These then also fall into a consistent relation with the insertion reactions of carbenes and the snap-out reactions of stable molecules from which latter, the present experimental evidence requires a large charge separation in the transition state. 

If observations are made at two points in space, it is possible to check the assumption g(z) = constant by comparing the initial distributions obtained from the two measured values. In general, if observations are made at n different space (or time) points, and an initial distribution g(z) is assumed with m arbitrary parameters, these parameters may be determined from consistency relations among the various observations, and there will be an additional n-m consistency relation to test the validity of the assumptions. Note that each test is actually multiplied by the number of particle size divisions, so that the initial distribution function q n be determined quite accurately if multiple observations are available. 

Here, coR is the frequency of motion in the reactant well, and Eb is the height of the transition-state barrier. Xr is the effective barrier frequency with which the reactant molecule passes, by diffusive Brownian motions through the barrier region and is given by the following self-consistent relation... 

It is obvious that the behavior at the Schwartzschild radius is different for the cases m0 A 0 and m0 = 0, see Eqs. (95), (103), and (104). Hence, we need to find consistent relations that link the two, independent of the specific eigenvalue of the mass operator examining the conjugate operator formulation based on the representations, i.e., Eqs. (98) and (104), where the energies scale as EopVl — 2k (r). By a straightforward generalization of Eq. (72), we obtain... 

We note that classical information is not distorted since the system operators r depend entirely on T2. However, from the consistency relations, Eqs. (G.6) and (G.7), we have established that T -> 0 as p, but also that Eqs. (G.5) and (G.7) break down at p = proving that IP is a Jordan matrix with Segr characteristic n = 2. Hence, we have proven that there is a singularity at the precise value p =, as well as the following theorem Theorem At the degenerate point p = the truth matrix T is singular, i.e., it is dissimilar to the zero matrix 0 since... 

However for a specific inflationary model, the four observable quantities As, At, ns and nj can be expressed at lowest order in term of the physical quantities 14 and the two slow-roll parameters < /,. and 5k, so that there exist some consistency relations (between the tensor spectral index and the scalar-to-tensor ratio), which in principle allow to test inflation, and to reconstruct the potential on a small region (as we can have access to V as well as V and V" with the slow roll parameters). Note however that this consistency relation crucially relies on the detection of the tensor modes (and hence, on the H-polarization of CMB as it is probably the most efficient way to detect the tensor modes), which may very well be an extraordinarily difficult task if /, happens to be very small. 

Unfortunately there are only limited data available on soil microbial populations that can utilize phenolic acids as a carbon source. The CFU/g soil ranged from 104 to 1010 for bacteria in phenolic acid enriched air-dried soil and not enriched field soils (Table 3.1). However, these numbers of active, quiescent, and/or dormant individuals represent only a small fraction of the bacteria in these soils, since <1% of the viable bacteria in soils can be cultured.43 The assumption is that the CFU based on cultures are representative or consistently related to the... 

If the coupling mechanism is dominated by exchange interaction, the correlation length Zex is self-consistently related to the total average anisotropy constant by... 

This method is absolutely analogous to one-site CPA on replacing the scalar quantities, such the self-energy, with nxn matrices. Therefore, one may show158 that the MCPA is also analytic and satisfies the necessary elementary conditions positivity of the spectrum, uniqueness of the solution, sum rules, etc. The self-consistency relation is obtained in a manner analogous to that of the one-site CPA (Section IV.A.4) it will be detailed in the calculation of the n x n matrix elements. 

Finally, we note that the following "classical consistency relations hold. For particles with a non-zero rest mass, the energy law gives, using Eqs. (32) and (33)... 

For multicomponent systems, the expression for y here employed may be shown equivalent to that involved in the cluster diagram technique (6), which is currently being employed in a variety of problems. The present derivation shows that the starting expressions satisfy the thermodynamic consistency relation embodied by the adsorption isotherm. It is, however, important to observe that any direct application of these alternative rigorous approaches, which is of necessity of an approximate nature, leads to some violation of the complete internal equilibrium conditions. Similarly, calculations of surface tension which employ the adsorption equation as a starting point invariably violate mechanical equilibrium in some order of approximation. 

The occupation probability (na) has to be determined from the self-consistency relation ... 

A consistent difference of -78% for pesticide use and environmental impacts between non transgenic- and transgenic, insect resistant- cotton is observed. This consistency relates to the fact that the same insecticides were assumed to have been applied to both non transgenic and transgenic cotton, the impact being caused by different numbers of insecticide sprays that had been applied to these crops. The consistent difference of -100% observed for the impacts of insect resistant maize relates to the feet that Gianessi et al. [7] assumed that this transgenic maize would provide 100% control of European com borer. 

This identification now makes for a completely consistent relation between the statistical properties of the gas and the thermodynamic varia-ables. The total energy of the gas, which is simply its kinetic energy for ideal monatomic gases, becomes [Eq. (VI.2.2)] for 1 mole... 

An approximate type of consistency relates to trends in properties within families in the periodic table. Expected trends can aid both in estimation of missing data and in evaluation of observed data, for example, in a series from fluoride to iodide or from polyhalide to monohalide. 

We see that the detailed balance is equivalent to Landauer s formula Eq. 3. The generic method of the steady state regime is utilized here in order to emphasize a common meaning of Landauer s ansatz of Eq. 3 corresponding to Eq. 15. The current flows and electronic distribution fdot(E) in the quantum dot are self-consistently related provided... 

Different Developers or Developers and Management Integrator tools help to assure fine-grained consistency relations between increments of different documents produced by different developers. Action patterns can also be used for actions of different persons. Both can be used for simultaneous as well as concurrent engineering. 

For subsequent checks on the MC simulations, from which we seek to determine Tjnvi it will prove convenient to derive a consistency relation that must hold regardless of molecular details of the specific model under r-onsideration. The derivation starts by assuming that an equation of staite t (T, A) (fixed TV, Sji) exists such that... 

Table 5.6 Compaiison of results for isostrain heat capacity from consistency relation Eq. (5.140) with directly computed values from canonical ensemble (CE) [see eq. (5.206)] for model A (see text). ...

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