Linearity calculation

Auxiliary subroutines for handling coordinate transformation between local and global systems, quadrature, convergence checking and updating of physical parameters in non-linear calculations. 

Secondly, as a measure of nonlinearity, the calculation conforms more closely to that concept than the correlation coefficient does. As a contrast, we can consider terms such as precision and accuracy, where high precision and high accuracy mean data with small values of standard deviation> while low precision and low accuracy mean large values of the measure. Thus, for those two characteristics, the measured value changes in opposition to the concept. If we were to use the correlation coefficient calculation as the measure of nonlinearity, we would have the same situation. However, by defining the linearity calculation the way did, the calculation now runs parallel to the concept a calculated value of zero means no nonlinearity while increasing values of the calculation corresponds to increasing nonlinearity. 

The data in Table 8.4 [4] represent the vapor pressure of mercury as a function of temperature. Plot In P as a function of 1/T to a scale consistent with the precision of the data. If the resultant plot is linear, calculate AH Iz from the slope obtained by a least-squares fit to the line. If the plot is curved, use a numerical differentiation procedure to obtain the value of AHmjZ as a function of T, and calculate ACpm- See Appendix A for methods. 

In the case of flows of polyethylene-based keroplast melts, the pressure profile along the channel length stayed linear. Calculations demonstrated that this is obviously explained by a relatively minor dependence of the composition viscosity on the temperature. The q value changes along the channel length by not more than two times. In a PE-keroplast flow in the low temperature gradient channels, pressure loss values are practically the same as in the isothermic flows. They can be calculated in compliance with the methods explained in Sect. 3. 

The calculated local impedance is presented in Figure 13.8 for Tafel kinetics with 7 = 1 and with radial position as a parameter. The impedance is largest at the center of the disk and smallest at the periphery, reflecting the greater accessibility of the periphery of the disk electrode. Similar results were also obtained for J = 0.1, but the differences between radial positions were much less sigiuficant. Inductive loops are observed at high frequencies, and these are seen in both Tafel and linear calculations for J = 0.1 and J = 1.0. ... 

In Fig. 4 we show results for a spin-boson system at low temperature and large Kondo parameter where the linearization approximation is expected to do most poorly. Indeed, in this situation we see a significant discrepancy between the results of our linearized calculations and exact values. The linearized path integral approximation overemphasizes the effect of the friction, and underestimates the importance of the coherent dynamics. Thus the exact result oscillates around zero while the linearized approximate result is overdamped and shows slow incoherent decay. 

We show in Fig. 2 a couple of examples for bare ions with Zj = 1 and The figure illustrates important differences between linear (or perturbative) and non-linear calculations. For Zj = 1 we note important quantitative differences at low and intermediate energies. The non-linear results exceed those of the linear theory by about 60% at low velocities. Thus, in the case of protons the non-linear theory predicts a significant enhancement of... 

On the other hand, in the case = 7 we observe a different behavior. Here the non-linear values lie well below those of the linear calculation on the whole velocity range. The interpretation of these results is also very revealing. The results obtained from the dielectric calculation increase by a whole factor 49 when going from = 1 to Zj = 7 - as expected from a linear formalism. Instead, the non-linear results increase by a much smaller factor (a ratio about 20 between both maxima). This may be interpreted as a saturation in the energy loss process, i.e., the values increase by a much... 

Another interesting result for energetic bare ions is illustrated in Fig. 3. Here we show the values of the stopping logarithm, obtained from the non-linear calculations of the energy loss, dE/dx, and using the relation L = — (d /dx)/(Zie(Up/v). The calculations correspond to an energy... 

Hence, in the high energy limit the non-linear calculations for bare ions reproduce the corresponding limits of the BBB theory. 

Fig. 4. Energy loss of ions (assuming frozen charge state conditions) as a function of the ion velocity r. The solid line is the result of the non-linear calculation for partially dressed ions (Z = 7, = 4). The dashed lines are the predictions of...

In Fig. 5 we show, with solid line, the non-linear calculations of the stopping power of carbon for all ions with atomic numbers in the range 1 < Zj < 40, and with a fixed velocity v = 0.8 a.u., together with experimental results from various authors [10,46,47]. We also show the theoretical results obtained from the DFT [32] (which, for the electron density of carbon targets, are available only in the range Zj 17), and the calculations according to the Brandt-Kitagawa model (BK) [15]. This model is based on linear theory and includes a statistical model for the ion stmcture as well as... 

Fig. 5. (a) Stopping power of carbon targets (with = 1-6) for slow ions with velocity v = 0.8 a.u. E/A = 16 keV/u). The solid line is the result of the present non-linear calculations for 1 < Zj <40 the dashed line shows the result of the density functional theory (DFT) for 1 < Zj < 17 (with = 1.5) the dotted line shows the result of the Brandt-Kitagawa model. The symbols show the experimental results from various authors [10,46,47]. (b) Contributions of the main partial wave components, I = 0,1,. ..,4, to the total stopping power shown in part (a). 

Fig. 9. General scaling of the effective charge values Z = [Sio (v)/5p(v)] /2 obtained from the non-linear calculations for all the ions with 1 < Zj < 92 in carbon targets, with energies E/M = 1, 2, 5, and lOMeV/u, assuming q = q. Here the calculated values are shown by symbols, while the solid line shows the empirical fitting to Zgff given by equation (19).

To understand the basic difference arising from the linear and non-linear models with respect to the deduced effective charge values, we have performed energy loss calculations for partially stripped ions with two models (i) the linear DF model, and (ii) the present non-linear theory (NL). The linear calculations have been performed using equation (6) and using the DF obtained by Lindhard [13]. The electronic structure of the ions in both calculations was represented by the same Moliere-ion function. 

For the moment, we describe the most naive approach to this problem and will reserve for later chapters (e.g. chap. 12) more sophisticated matching schemes which allow for the nonlinear calculations demanded in the core to be matched to linear calculations in the far fields. As we have repeatedly belabored, the objective is to allow the core geometry to emerge as a result of the full nonlinearity that accompanies that use of an atomistic approach to the total energy. However, in order to accomplish this aim, some form of boundary condition must be instituted. One of the most common such schemes is to assume that the far field atomic positions are dictated entirely by the linear elastic fields. In particular, that is... 

It is at present too soon to judge the usefulness of curvilinear coordinates for coplanar and spatial motions. From the numerical results and analytical developments published so far, it is clear that generalizations to a plane or space have been possible only through a large increase of effort over col-linear calculations, due to the increased complexity of the equations to be solved. However, this is a usual situation in the preliminary stages of most work and may change in the future. Another open question, that will have to be answered, is the extent to which curvilinear coordinates will be useful in situations where surface hopping , i.e. electronic transitions, are involved. 

As an example, a PtlOO resistor has 138.5 Q at 100 °C. Based upon the linear calculated thermal coefficient at the 100 °C and 0°C points... 

Non-linear least squares calculations are more complex than linear calculations. They start with an initial estimate and find the minimum variance by using a sequence of iterations. It is possible, and common, to converge upon local minima rather than the global minimum Numerical least squares techniques are described in [88-prefla]. 

Linearity. Calculate and the y intercept as a percentage of the midrange response. Calculate the response factor (RF) for each experimental point on the line, and using Excel, plot the RF vs. concentration. From the slope (RF/unit concentration), calculate the RF change over the range of experimental points and calculate this as a percentage of the average RF value. 

All of the above conclusions were based on the linearized equations for small perturbations about the steady state. A theorem of differential equations states that if the linearized calculations show stability, then the nonlinear equations will also be stable for sufficiently small perturbations. For larger excursions, the linearizations are no longer valid, and the only recourse is to (numerically) solve the complete equations. A definitive study was performed by Uppal, Ray, and Poore [40] where extensive calculations formed the basis for a detailed mathematical classification of the many various behavior patterns possible refer to the original work for the extremely complex results. The evolution of multiple steady states when the mean holding time is varied leads to even more bizarre possible behavior (see Uppal, Ray, and Poore [41], Further aspects can be found in the comprehensive review of Schmitz [42] and in Aris [1], Perlmutter [31], and Denn [43]. 

Measurements were carried out to verify the non-linear calculation methods of longitudinal forces in long, high railway bridges. The practical and theoretical background of the tests is described. Specially chosen methods of loading and measuring the reaction of the structure are appointed to the separate static and dynamic problems. 

Run conditions (l-C8Hi6l/[C2HsOH]/[PdJ = 100 100 1 100 atm 80°C 8 hr A mixture of ethyl nonanoate with some ethyl 2-methyloctanoate and ethyl 2-ethylheptanoate. j Ester linearity calculated basis ethyl nonanoate/total linear plus branched Cj-acid ester. Total linear plus branched Cj-acid ester yield, calculated basis 1-octene charged. 

Linearity calculated basis linear alcohol or aldehyde/total alcohol or aldehyde, r Yield calculated basis olefin converted. 

The various linear calculations show that two axisymmetric colloids on a membrane should repel. But, as the detachment angles a,- increase, it becomes harder to justify the linearization. The expansion in Eq. (3) ultimately rests on the smallness of IV/il, an expression that should be compared to tan a,. But, once higher order terms matter, Monge parametrization not only becomes technically impenetrable, it is even incapable of dealing with membrane shapes that display overhangs. It is hence preferable to discard it in favor of a more general numerical surface triangulation. 

In some special analyses it may be necessary to extend the absorbance measurements beyond the linear range (i.e., beyond the validity of Beer-Lamberts s law). In these cases a non-linear calibration curve has to be generated by measuring concentrations versus spectrophotometer readings. The sample concentrations are then calculated according to the non-linear calibration function or corresponding correction terms applied to the linearly calculated sample concentrations. 

---