Null coefficients

Ishikawa. T, Fukunaga, H. and Ono, K.I. (1989) Graphite-epoxy laminates with almost null coefficient of thermal expansion under a wide range of temperature. J. Mater. Sci. 24,2011-2017. 

Reduce the number of MTP coefficients on each site In Section 7.2.2, we argued in favor of aligning reference-axis systems along symmetries of an atom s immediate environment, which leads to a number of MTP coefficients to being zero. By computing energies and forces/torques in the MTP sites local axis systems, these null coefficients cancel all interactions in which they participate. Given... 

It is worth noting that, if we look at an expansion with two non-null coefficients and 5, we can write, for component 1, for instance ... 

Null (1970) discusses some alternate models for the excess Gibbs energy which appear to be well suited for systems whose activity coefficients show extrema. 

The activity coefficient y can be defined as the escaping tendency of a component relative to Raonlt s law in vapor-liqnid eqnihbrinm (see Sec. 4 in this handbook or Null, Phase Equilibrium in Process Design, Wiley-Interscience, 1970). 

The methods of simple and of inverse iteration apply to arbitrary matrices, but many steps may be required to obtain sufficiently good convergence. It is, therefore, desirable to replace A, if possible, by a matrix that is similar (having the same roots) but having as many zeros as are reasonably obtainable in order that each step of the iteration require as few computations as possible. At the extreme, the characteristic polynomial itself could be obtained, but this is not necessarily advisable. The nature of the disadvantage can perhaps be made understandable from the following observation in the case of a full matrix, having no null elements, the n roots are functions of the n2 elements. They are also functions of the n coefficients of the characteristic equation, and cannot be expressed as functions of a smaller number of variables. It is to be expected, therefore, that they... 

Figure 2. Right figure amplitude of the Miju coefficients of a real mirror. If the mirror were perfect, then only diagonal elements would be non-null. Left figure surface figure map of a Virgo mirror.

Z is a coefficient which relates the concentration of the analyte in the unknown sample to the concentration in the calibration standard, where = bc. R is a residual matrix which contains the measurement error. Its rows represent null spectra. However, in the presence of other (interfering) compounds, the residual matrix R is not random, but contains structure. Therefore the rank of R is greater than zero. A PCA of R, after retaining the significant PCs, gives ... 

Several equations have been developed to represent the dependence of activity coefficients on liquid composition. Only those of most use in the design of separation processes will be given. For a detailed discussion of the equations for activity coefficients and their relative merits the reader is referred to the book by Reid et al. (1987), Walas (1984) and Null (1970). 

Relatively simple experimental techniques, using ebulliometry and chromatography, are available for the determination of the activity coefficients at infinite dilution. The methods used are described by Null (1970) and Conder and Young (1979). 

For systems that are only partially miscible in the liquid state, the activity coefficient in the homogeneous region can be calculated from experimental values of the mutual solubility limits. The methods used are described by Reid et al. (1987), Treybal (1963), Brian (1965) and Null (1970). Treybal (1963) has shown that the Van-Laar equation should be used for predicting activity coefficients from mutual solubility limits. 

The null hypothesis test for this problem is stated as follows are two correlation coefficients rx and r2 statistically the same (i.e., rx = r2)l The alternative hypothesis is then rj r2. If the absolute value of the test statistic Z(n) is greater than the absolute value of the z-statistic, then the null hypothesis is rejected and the alternative hypothesis accepted - there is a significant difference between rx and r2. If the absolute value of Z(n) is less than the z-statistic, then the null hypothesis is accepted and the alternative hypothesis is rejected, thus there is not a significant difference between rx and r2. Let us look at a standard example again (equation 60-22). 

And Z(n) = 0.89833, therefore Z(n), the test statistic, is less than 1.96, the z-statistic, and the null hypothesis is accepted - there is not a significant difference between the correlation coefficients. 

As noted earlier, the sum of the mass fractions is unity and thus Eq. (86) will be consistent with Eq. (85) only if the sum of the correction term b m over all chemical species a = 1,..., K is null. In general, this will not be the case if Eq. (89) is used. Another difficulty that can arise is that the mass fractions in two environments may be equal, e.g., i = a2, and thus the coefficient matrix in Eq. (89) will be singular. This can occur, for example, in the equilibrium-chemistry limit where the compositions depend only on the mixture fraction, i.e., (j) — co(0- F°r chemical species that are not present in the feed streams, the equilibrium values for = 0 and = 1 are zero, but for intermediate values of the mixture fraction, the equilibrium values are positive. This implies that the equilibrium values will be the same for at least two values of the mixture fraction in the range 0< < 1. Thus, in the equilibrium limit it is inevitable that two environments will have equal mass fractions for certain species at some point in the flow field. Since singularity implies an underlying correlation between... 

These coefficients have been obtained from the eigenvectors associated with the null eigenvalues of the mineralogical matrices formed out of 6 end-members or minerals at a time. No more than three equations from this set can be considered independent. 

The canonical correlation coefficients can also be used for hypothesis testing. The most important test is a test for uncorrelatedness of the x- and y-variables. This corresponds to testing the null hypothesis that the theoretical covariance matrix between the x- and y-variables is a zero matrix (of dimension mx x mY). Under the assumption of multivariate normal distribution, the test statistic... 

The Hausman test was used to test the null hypothesis that the coefficients estimated by the efficient random-effect model are the same as the ones estimated by the consistent fixed-effect model. If this null hypothesis cannot be rejected (insignificant P-value in general, it is larger than 0.05), then the random-effect model is more appropriate. 

Figures 3 and 4 display the paired results obtained with Ab-cor and 3M diffusion-type samplers, respectively. These results were analyzed through use of the "t" test for paired samples and the calculation of correlation coefficients and regression equations, with the results of these analyses shown in Table I. A statistically significant correlation is seen between the data set for each type of diffusion sampler and the corresponding tube/ pump sample data set, and the "t" test fails to refute the null hypothesis that there is no significant systematic difference between each of the diffusion sampler data set and the corresponding tube/pump data set.

D Dj Do, DR E Ea E° Ea Ec Em, Eoui m-3) diffusion coefficient (m2 s 1) diffusion coefficient of species (m2 s-1) diffusion coefficient of species O and R (m2 s-1) electrode potential (vs. some reference electrode) (V) null or equilibrium potential (V) standard potential (V) interelectrode potential (V) voltage input and output of a circuit (V)... 

Remark 1. If all the atomic coefficients /3 are positive we speak (where necessary) of a proper molecular species. If all the /3( are zero we have the null or trivial species, and if any are negative a pseudo-species. ... 

Corporation attempted to measure the B3 field. However, the results were null, and an inconclusive direct measurements of the B3 field still remains elusive. On the theoretical front non-Abelian electrodynamics remains controversial and not widely upheld. Some objections are not entirely reasonable. On the other hand, Waldyr Rodriques objected to certain assumptions, proposed by M. W. Evans, that relates coefficients in Whittaker s 1904 paper on electrodynamics to the putative existence of longitudinal modes in non-Abelian electromagnetic waves in vacuum. Rodrigues objections appear reasonable. However, this response was quite forceful and direct, and resulted in his refusal to consider anything involving non-Abelian electrodynamics. 

In Fig. 3.14a, the dimensionless limiting current 7j ne(t)/7j ne(tp) (where lp is the total duration of the potential step) at a planar electrode is plotted versus 1 / ft under the Butler-Volmer (solid line) and Marcus-Hush (dashed lines) treatments for a fully irreversible process with k° = 10 4 cm s 1, where the differences between both models are more apparent according to the above discussion. Regarding the BV model, a unique curve is predicted independently of the electrode kinetics with a slope unity and a null intercept. With respect to the MH model, for typical values of the reorganization energy (X = 0.5 — 1 eV, A 20 — 40 [4]), the variation of the limiting current with time compares well with that predicted by Butler-Volmer kinetics. On the other hand, for small X values (A < 20) and short times, differences between the BV and MH results are observed such that the current expected with the MH model is smaller. In addition, a nonlinear dependence of 7 1 e(fp) with 1 / /l i s predicted, and any attempt at linearization would result in poor correlation coefficient and a slope smaller than unity and non-null intercept. 

In order to establish the most appropriate conditions for the determination of the diffusion coefficients of both electroactive species by using Eqs. (4.38) and (4.39), it has been reported that when the reaction product is absent (i.e., cR = 0) neither planar electrodes nor ultramicroelectrodes can be used in DPC for determining diffusion coefficients (see Eqs. (4.48) and (4.50)) because in these situations the anodic limiting current is either independent of DR or null, respectively. 

The calculated value of analysis of variance is F=1343.6 for the null hypothesis HqiP O. However, since the tabular value is F1 g 0 95=5.32 the null hypothesis is rejected and the alternative hypothesis accepted that the regression coefficient p, with 95% confidence level is statistically significant. 

Variation levels Null level Upper level Lower level +1.41 -1.41 Factors Regression coefficients ... 

N-is number of trials included in determining regression coefficients k-is number of coefficients in regression equation n0-is number of replicate trials in null point. 

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