Quadratic curve

Parabola or quadratic curve Cubic curve nth degree curve... 

Weighting of the calibration curve, 1 /x or 1 /x, is expected to provide better curve fit at the lower concentration levels. Alternative calculations, such as exponential or quadratic curve fits, are acceptable if they provide improved precision and/or accuracy. 

The solid lines in Fig. 10 represent the quadratic curves obtained in regression analyses. 

In accordance with Eq. (48), the values of ACiJ (z-dep)/(47rr ) have been plotted against E in Fig. 11. The respective plots for the three ion groups have been found to lie on a single quadratic curve. 

The method of least squares permits us to calculate the best function of a given form for the set of data at hand, but it does not help us decide which form of analytic function to choose. Inspection of a graph of the data is helpful in such a choice. Figure A.l shows the data of Table A.l as well as the best straight line and the best quadratic curve, with the latter represented by Equation (A. 12), both are fitted to the data by the method of least squares. 

Figure A.l. A plot of the data of Table A.l, and the best linear and quadratic curves fitted to the data by the method of linear least squares.

The second-order rate constants for thiocyanate anation vs pH are shown in Fig. 1.13. The full line represents (1.216) with the values shown in scheme (1.217). This profile had been earlier recognized in the ring closure of the three analogous pH-related forms of Co(III)-edta to give Co(edta) in which the edta is completely coordinated.In the Co(lll) case the reactivities of the three forms are much closer. A plot of A [H+] -1-[H+] is a quadratic curve from which / ah2> ah be obtained. 

Considering now the calculated energies of the truncated expansions as functions of the estimated normalization deficiencies, one finds that, in all cases, the energies approach the flill-SDTQ value for the limit Ac (Ntr)-- 0 from above along a linear or weakly quadratic curve. By means of an extrapolation of this curve one can then determine the degree of truncation necessary for the error not to exceed the desired threshold, say 1 mh, without having to calculate a Cl wavefimctions larger than the truncated one. 

The dependences of wave velocity on the ratio of cubic to quadratic processes, described by eqns (11.43) and (11.44) are shown in Fig. 11.6. The quadratic curve always lies below the cubic result, but the two are tangential at q = 1/2, c = yfl. Numerical computations reveal the following detail concerning the stability of the respective solutions. For q < i, where cubic dominates quadratic, it is the velocity determined by eqn (11.43) which is selected by the system. Above q = j, however, the quadratic character takes over, and the minimal velocity described by eqn (11.44) emerges dominant. 

Figure 4-11 Calibration curve for protein analysis in Table 4-7. The equation of the solid straight line fitting the 14 data points (open circles) from 0 to 20 y.g, derived by the method of least squares, is y = 0.016 3 ( 0.000 22)x t-0.004, (+0.0024). The standard deviation of y is sy = 0.0059. The equation of the dashed quadratic curve that fits all 17 data points from...

If using a computer program, use a quadratic curve fit for the nonlinear standard curve to calculate the protein concentration of the samples. If the standard curve is linear or if the absorbance readings for the samples fall within the linear portion of the standard curve, the total protein concentrations of the samples can be estimated using the linear regression equation. 

Fig. 2.2 Actual molecules do not sit still at the bottom of the potential energy curve, but instead occupy vibrational levels. Also, only near qe, the equilibrium bond length, does the quadratic curve approximate the true potential energy curve...

The more steeply rising curve is the longitudinal or compression mode. The less steep curve at — 0 is the torsional mode. The quadratic curves at 9 = 36 are beam bending modes. The dashed curve is the dispersion obtained after taking into account the coupling to water. 

The relationship of the thermal conductivities of fabrics and volume fractions of water in the interfiber spaces was expressed by a quadratic curve when the heat flow was normal to the fabric surface and by a straight line when the flow was parallel to the warp yarns. Except for hairy wool fabrics, the thermal conductivity of various wet fabrics may be calculated from the equations of Naka and Kamata (J3). An earlier investigation used an environmentally controlled room as a periodic heat source, and observed conductivities of 1-2 x 10 l cal/cm-sec °C for cotton, linen, and wool fabrics, and changes to 2-10 x 10 when the water content of these fabrics were increased ( ). After correcting for anisotropic effects, good agreement between actual conductivity measurements of wool fabrics and those calculated from a mathematical model of a random arrangement of fibers was observed. 

A quadratic function defines a symmetric parabola and therefore cannot exactly reproduce the true relationship between the distortion of a bond length or valence angle and the energy needed to effect that distortion. However, a central assumption in the application of simple molecular mechanics models is that distortions from ideal values are small and in such cases it is only necessary that the potential energy function be realistic in the region of the ideal value. This is shown in Fig. 17.8.1, where a quadratic curve is compared to a Morse potential that is believed to more accurately reflect the relationship between bond length distortion and energy cost. 

Figure 7. The least-square fit quadratic curves relating the CC bond distances in the D3/, distorted benzene to the average s-character (a) and the 7r-bond order (b).

Application of a computer program to these data of Fig. 24 yielded calculated values of a for every possible solute pair in the seven mobile phases used. The program then interpolated these data over the entire compositional triangle by fitting to a quadratic curve, to yield values of a as a function of mobile-phase composition. Finally, these a values are plotted in trilinear form (Fig. 25) in such a manner as to indicate mobile-phase compositions of optimum selectivity. Figure 25a shows such a plot for band pair 6-8, where the white region indicates resolution of the two bands (on one 25-cm silica column) greater than the minimum desired Rs > 1.0). 

For a quadratic curve fitted to the data, the model can be expressed as... 

The method of least squares was employed in the previous section to fit the best straight line to analytical data and a similar procedure can be adopted to estimate the best polynomial line. To illustrate the technique, the least squares fit for a quadratic curve will be developed. This can be readily extended to higher power functions. - ... 

Orthogonal polynomials are particularly useful when the order of the equation is not known beforehand. The problem of finding the lowest-order polynomial to represent the data adequately can be achieved by first fitting a straight line, then a quadratic curve, then a cubic, and so on. At each stage it is only necessary to determine one additional parameter and apply the f-test to estimate the significance of each additional term. 

Figure 3.14 Hydraulic resistance A = 2/ of a duct with EPR vs Reynolds number Re and EPR density A for linear (a) quadratic (curves b) force laws 1-4=1,2-10 and 3 - 20 EPR height h = 0.3.

As long as the range of the interaction and the separation h is much less than the radius of curvature of the system, it is a valid approximation for interaction between surfaces quadratically curved in the vicinity of the point of closest approach. The condition makes curvature effects, higher than second order (via truncation of a Taylor series expansion), on the approximated energy significantly small. However, the... 

The slope is given in the unconventional units, cm (A ) . (Conventionally the momentum transfer in Fig. 5.9 would be linear in Q and the lines of equivalent mass become quadratic curves (see Fig. 9.18).)... 

Vibrational frequencies are calculated to obtain IR spectra, to characterize stationary points, and to obtain zero point energies (below). The calcnlation of meaningful frequencies is valid only at a stationary point and only using the same method that was used to optimize to that stationary point (e.g. an ab initio method with a particular correlation level and basis set - see chapter 5). This is because (1) the use of second derivatives as force constants presupposes that the PES is quadratically curved along each geometric coordinate q (Eig. 2.2) but it is only near a stationary point that this is true, and (2) use of a method other than that nsed to obtain the stationary point presupposes that the PES s of the two methods are parallel (that they have the same... 

Hence the valve pressure ratio at choking may be calculated from the quadratic curve... 

Fig- 5. Results of the quadratic curve fitting for the reduced spreading coefficient vs fractional polarity for HPMC, MC and PVP [69]... 

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