Two parameters

Since the accuracy of experimental data is frequently not high, and since experimental data are hardly ever plentiful, it is important to reduce the available data with care using a suitable statistical method and using a model for the excess Gibbs energy which contains only a minimum of binary parameters. Rarely are experimental data of sufficient quality and quantity to justify more than three binary parameters and, all too often, the data justify no more than two such parameters. When data sources (5) or (6) or (7) are used alone, it is not possible to use a three- (or more)-parameter model without making additional arbitrary assumptions. For typical engineering calculations, therefore, it is desirable to use a two-parameter model such as UNIQUAC. 

A high degree of correlation may be beneficial. When the parameters are strongly related, some linear combination of the two parameters may represent the data as well as do the individual parameters. In that case a method similar to that of Bruin and Praus-... 

We use Equation (2) primarily with five parameters, or with four parameters, excluding C. When data were sparse or of poor precision, a linear two-parameter fit (C = = 0) was... 

If only two parameters are estimated, C must be set to some value. 

If only two parameters are fit, C must set to some arbitrary value, usually one, and only Pqj and P 2) estimated from the VLE... 

If only two parameters are fit, a must be set to some predetermined value, and only and P 2) determined from the binary... 

This is the essential characteristic for every lubricant. The kinematic viscosity is most often measured by recording the time needed for the oil to flow down a calibrated capillary tube. The viscosity varies with the pressure but the influence of temperature is much greater it decreases rapidly with an increase in temperature and there is abundant literature concerning the equations and graphs relating these two parameters. One can cite in particular the ASTM D 341 standard. 

Fig.5 Weibull two-parameters dividing. Exemple of results for grey cast iron with heat treated surfaces [5],...

In the work presented here, a slightly different two-parameter transient model has been used. Instead of specifying a center frequency b and the bandwidth parameter a of the amplitude function A(t) = 6 , a simple band pass signal with lower and upper cut off frequencies and fup was employed. This implicitly defined a center frequency / and amplitude function A t). An example of a transient prototype both in the time and frequency domain is found in Figure 1. 

In the systematic approach, the contaminated signal was processed using transients with parameters selected from a uniformly sampled grid in the parameter space. For each parameter value, the quality of the processed signal was computed. An example result is presented in Figure 2 which shows the performance as a function of the two parameters and / p. The parameter values /, and which yielded the lowest entropy were selected for processing. 

Of course, it is well known that because of the duality of the time and frequency parameters, there is no universal time-frequency technique. Indeed the two parameters cannot be simultaneously known with arbitrarily high resolution. This is expressed by the Heisenberg - Gabor inequality ... 

The desilvering speed of an electrolytic silver recovery unit can be characterized by two parameters. 

The second parameter is the response. The response relates to the silver levels needed to reach a certain current. The two parameters are shown in Figure 1. The response corresponds to the... 

Figure Al.6.30. (a) Two pulse sequence used in the Tannor-Rice pump-dump scheme, (b) The Husuni time-frequency distribution corresponding to the two pump sequence in (a), constmcted by taking the overlap of the pulse sequence with a two-parameter family of Gaussians, characterized by different centres in time and carrier frequency, and plotting the overlap as a fiinction of these two parameters. Note that the Husimi distribution allows one to visualize both the time delay and the frequency offset of pump and dump simultaneously (after [52a]).

In this expression, the packing fraction r] = Ttpa- /6, and the other two parameters are related to this by... 

The brackets symbolize fiinction of, not multiplication.) Smce there are only two parameters, and a, in this expression, the homogeneity assumption means that all four exponents a, p, y and S must be fiinctions of these two hence the inequalities in section A2.5.4.5(e) must be equalities. Equations for the various other thennodynamic quantities, in particular the singidar part of the heat capacity Cy and the isothemial compressibility Kp may be derived from this equation for p. The behaviour of these quantities as tire critical point is approached can be satisfied only if... 

Unfortimately for modem crystallographers, all of tlie crystal stmctiires that could be solved by the choose-the-best-of-a-small-niunber-of-possibilities procedure had been solved by 1920. Bragg has been quoted as saying that the pyrite stmcture was very complicated , but he wrote, in about 1930, It must be realized, however, that (cases having one or two parameters) are still extremely simple. The more typical crystal may have ten, twenty, or forty parameters, to all of which values must be assigned before the analysis of the stmcture is complete. This statement is read with amusement by a modem crystallographer, who routinely works with hundreds and frequently with thousands of parameters. 

Two parameters must be measured to apply the BET equation, the pressure at the sample and the amount adsorbed at this pressure. There are tlnee conmron methods for measuring the amount of gas adsorbed, called the volumetric method, the gravimetric method and the dynamic method, of which the volumetric method is the connnonest [21],... 

In this case, only two parameters (k and Iq) per atom pair are needed, and the computation of a quadratic function is less expensive. Therefore, this type of expression is used especially by biomolecular force fields (AMBER, CHARMM, GROMOS) dealing with large molecules like proteins, lipids, or DNA. 

For every type of angle including three atoms, two parameters (force constant fe and reference value 0q) are needed. Also, as in the bond deformation case, higher-order contributions such as that given by Eq. (23) are necessary to increase accuracy or to account for larger deformations, which no longer follow a simple harmonic potential. 

Each combination of four atoms (A, B. C. and D) is characterized by two parameters, e and e.. As for the CICC, is a parameter that depends on atomic properties and on distances, and is calculated by Eq. (27), with r, again being the sum of bond lengths between atoms on the path with the minimum number of bond counts. However c is now a geometric parameter (dependent on the conformation)... 

You need to specify two parameters the et uilibrium value ofthe internal coordinate and the force constant for the harmonic poten tial, T h e equilibrium restraint value deperi ds on the reason you choosea restraint. If, for example, you would like a particular bond length to remain constant during a simulation, then the equ ilibritirn restrain t value would probably be Lh e initial len gth of the bond. If you wan t to force an internal coordinate to a new value, the equilibrium internal coordinate is the new value. 

A Gaussian expansion contains two parameters the coefficient and the exponent. The most flexible way to use Gaussian functions in ab initio molecular orbital calculations permits both of these parameters to vary during the calculation. Such a calculation is said to use... 

The orbital exponent 4 is expressed as a function of two parameters a and / as follows ... 

Hential is an exponential-6 potential with just two parameters the mirrimum energy 5 and the well depth e [Hill 1948] ... 

II energy values where the energy is between Ef j and E and where the value of the opriate array element H( ) is greater than some minimum value (e.g. 20). In other is, that particular energy level must have been visited at least twenty times during iimulation. The following two parameters are now calculated ... 

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