Linear fit

Figure A2.5.25. Coexistence-curve diameters as functions of reduced temperature for Ne, N2, C2H4, and SFg. Dashed lines indicate linear fits to the data far from the critical point. Reproduced from [19] Pestak M W, Goldstein R E, Chan M H W, de Bniyn J R, Balzarini D A and Ashcroft N W 1987 Phys. Rev. B 36 599, figure 3. Copyright (1987) by the American Physical Society.

Beyond the CMC, surfactants which are added to the solution thus form micelles which are in equilibrium with the free surfactants. This explains why Xi and level off at that concentration. Note that even though it is called critical, the CMC is not related to a phase transition. Therefore, it is not defined unambiguously. In the simulations, some authors identify it with the concentration where more than half of the surfactants are assembled into aggregates [114] others determine the intersection point of linear fits to the low concentration and the high concentration regime, either plotting the free surfactant concentration vs the total surfactant concentration [115], or plotting the surfactant chemical potential vs ln( ) [119]. 

Conductivity measurements yield molar conductivities A (Scm2 mol-1) at salt concentration c (mol L-1). A set of data pairs (Af, c,), is evaluated with the help of non linear fits of equations [89,93,94] consisting of the conductivity equation, Eq. (7), the expression for the association constant, Eq. (3), and an equation for the activity coefficient of the free ions in the solution, Eq.(8) the activity coefficient of the ion pair is neglected at low concentrations. 

Thus, the enhancement of heat transfer may be connected to the decrease in the surface tension value at low surfactant concentration. In such a system of coordinates, the effect of the surface tension on excess heat transfer (/z — /zw)/ (/ max — w) may be presented as the linear fit of the value C/Cq. On the other hand, the decrease in heat transfer at higher surfactant concentration may be related to the increased viscosity. Unfortunately, we did not find surfactant viscosity data in the other studies. However, we can assume that the effect of viscosity on heat transfer at surfactant boiling becomes negligible at low concentration of surfactant only. The surface tension of a rapidly extending interface in surfactant solution may be different from the static value, because the surfactant component cannot diffuse to the absorber layer promptly. This may result in an interfacial flow driven by the surface tension gradi-... 

Fig. 13. Plot of the g values g,gz) and of the average g value vs rhombicity (Uj) of the Rieske and Rieske-type proteins listed in Table VIII. The lines represent linear fits to the data points.

Fig. 14. Plot of the g values g,g ) and of the average g value g vs rhombicity (UJ of (a) wild type (open symbol) and variant forms (closed symbols) of the Rieske protein in yeast bci complex where the residues Ser 183 and Tyr 185 forming hydrogen bonds into the cluster have been replaced by site-directed mutagenesis [Denke et al. (35) Merbitz-Zahradnik, T. Link, T. A., manuscript in preparation] and of (b) the Rieske cluster in membranes of Rhodobacter capsulatus in different redox states of the quinone pool and with inhibitors added [data from Ding et al. (79)]. The solid lines represent linear fits to the data points the dashed lines reproduce the fits to the g values of all Rieske and Rieske-type proteins shown in Fig. 13.

Dependence of measured mass burning flux on unburned mixture temperature for (a) ethylene/air, (b) n-heptane/air and iso-octane/air, and (c) n-decane/air and n-dodecane/air mixtures. The dotted lines represent the linear fits. 

Equation 15 represents the best linear fit to the data and it was calculated using program GPCl. 

Figure 4. Cross-sectional bright-field TEM views of Au-implanted silica samples at 3 x lO Au /cm, 190 keV, aimealed for 1 h at (a) 400 °C in air, (b) 700 °C in air, (c) 900 °C in air, and (d) 900 °C in Ar, respectively (e) the histograms of the size distribution of the samples annealed 1 h in air at different temperatures (f) Arrhenius plot of the squared average cluster radius after 1 h annealing in air (filled circles) or argon (empty triangles). Solid lines are linear fit to the experimental data.

Table 5.4 Linear fit data for Fe Mossbauer isomer shift predictions using the linear equation 5 = h (p - c) + a. A collection of 21 iron complexes with varying charge, oxidation- and spin-states have been studied (taken from [11])...

Fig. 6.15 Induced magnetic hyperfine fields, estimated from the spectra in Fig. 6.14, as a function of the reciprocal applied magnetic field. The full lines are linear fits in accordance with (6.20). The dotted line is a fit to the Langevin function. (Reprinted with permission from [58] copyright 1985 by the American Chemical Society)...

Fig. 6.17 The average magnetic hyperfine field at low temperatures, obtained from Mossbauer spectra of coated and uncoated 8 nm particles of a-Fe203 (Fig. 6.16). The lines are linear fits to the data in accordance with (6.23) and (6.25). (Reprinted with permission from [77] copyright 2006 by the American Physical Society)...

The equation by Hamaker is one of the most commonly used methods for describing dissipation kinetics using a linear fit. The basic computational form of the equation is... 

As mentioned previously, most agrochemicals do not exhibit linear degradation patterns. As a result, Hamaker proposed another variation of the linear-fit equation that allows better description of nonlinear data sets ... 

The first order plot for the isomerization reaction shows a good linear fit (correlation coefficient = 0.998) while there is... 

Figure 3.13 Reaction rates, determined from the change in the size of the (2 x 2) area between successive panels of the data of Figure 3.12, normalized to (squares) the length of the boundary between oxygen and CO domains (the full line is a linear fit) and...

When the VFT equation was applied every system gave a linear fit (Fig. 9). This implies that ion transport was assisted by segmental motion of polymer chains in a... 

Juez and Tamayo51 also apply time-series analysis to the evaluation of the consequences of introducing selective financing in 1993. Using the aggregate monthly data on pharmaceutical expenditure of the National Health System between 1991 and 1995, in constant deseasonalized pesetas, the authors compare the observed evolution with the theoretical evolution according to a linear fit. They conclude that the measure had a notable effect in the short term, but was absorbed in the long term. 

As an example, we applied these concepts to the Anscombe data [7], Table 66-1 shows the results of applying this to both the normal data (Anscombe s XI, Y1 set) and the data showing nonlinearity. We also computed the nature of the fit using only a straight-line (linear) fit as was done originally by Anscombe and also fitted a polynomial using the quadratic term as well. It is interesting to compare results both ways. 

X-axis. It presents the coefficients of the linear models (straight lines) fitted to the several curves of Figure 67-1, the coefficients of the quadratic model, the sum-of-squares of the differences between the fitted points from the two models, and the ratio of the sum-of-squares of the differences to the sum-of-squares of the X-data itself, which, as we said above, is the measure of nonlinearity. Table 67-1 also shows the value of the correlation coefficient between the linear fit and the quadratic fit to the data, and the square of the correlation coefficient. 

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