Curvilinear approach

These effects are illustrated in a comparison of the rectilinear and curvilinear approaches to differential sedimentation presented in Figure 8, adapted from Han and Lawler (24). In both approaches, the upper, larger, faster particle is settling by gravity toward the lower, smaller, slower particle. In the rectilinear or Smoluchowski approach, all small particles with size di that reside below the larger particle (size dp within the area Ar with diameter (d, + dj) come into contact with the larger particle ( DS(ij) is given by equation 3c. 

If the inhibition is found to be rapidly reversible, we must next determine if the approach to equilibrium for the enzyme-inhibitor complex is also rapid. As described in Chapter 4, some inhibitors bind slowly to their target enzymes, on a time scale that is long in comparision to the time scale of the reaction velocity measurement. The effect of such slow binding inhibition is to convert the linear progress curve seen in the absence of inhibitor to a curvilinear function (Figure 5.10). When nonlinear progress curves are observed in the presence of inhibitor, the analysis of... 

Even when using LDPE and PP, the diffusive losses of most nonpolar solvents may be unacceptably large for A designs that exceeded 1 cm . This is especially true at higher exposure temperatures because both solute diffusion and polymer free volume increase with temperature (Comyn, 1985). Even when solvent losses were not excessive, uptake of HOCs by membrane-enclosed solvents appeared to become curvilinear well before thermodynamic equilibrium was approached. This phenomenon is likely due to the outward flux of sampler solvent with elevated HOC levels (relative to water), which appears to facilitate residue... 

The binding process may exert a profound effect on cell function. It is well known that hepatic cells do not function when cultured on a flat plate. This reason may be partially explained by the binding-spreading phenomenon. It therefore follows that an ideal scaffold would minimize the spreading effect. Inasmuch as hepatic cells are our primary interest, a scaffold that best approaches the native shapes of the cells would appear to be the best. It is equally clear that this shape must be optimized against additional requirements of void volume and mass transport. Nevertheless, an ideal scaffold would be more curvilinear than flat. 

One category of finite-difference method uses a rectangular grid. In this approach one covers the specified layout with a grid, or mesh, as shown in Figure i 15.2a. When curvilinear boundaries are involved, it is possible to sample the... 

Young et al. [43] chose an alternative approach to prediction of ultimate pH. They measured the concentration of the muscle glycogen responsible for lactic acid formation. This was done immediately after slaughter. The glycogen concentration, expressed as glucose equivalents per gram of muscle, has a curvilinear relationship to ultimate pH (Fig. 28.1). 

Curvilinear regression should not be confused with the nonlinear regression methods used to estimate model parameters expressed in a nonlinear form. For example, the model parameters a and b in y = axb cannot be estimated by a linear least-squares algorithm. Information in Chapter 7 describes nonlinear approaches to use in this case. Alternatively, a transformation to a linear model can sometimes be used. Implementing a logarithmic transformation on yt = ax/ produces the model log y = log a + blog x which can now be utilized with a linear least-squares algorithm. The literature [4, 5] should be consulted for additional information on linear transformations. 

When plotted as -SC versus C, Equation 4.15 produces a curvilinear line approaching asymptotically ST (Fig. 4.20). When KLC < < < 1 (Eq. 4.15a), a plot of -SC versus C would produce a straight line with slope KLST and the y intercept would be zero. Upon rearranging Equation 4.15 such that... 

A plot of ExCa versus CRCa will produce a curvilinear line, asymptotically approaching CEC. The pathway of such line from ExCa = 0 to ExCa = CEC depends on ATCa Mg and CEC (see the section entitled Relationship Between (CR, and ExCa). 

The limits of Equations B and C at zero and infinity imply that a plot of Equation B in terms of CRCa versus ExCa (with ExCa being the dependent variable) at constant CEC and ACa Mg will give a curvilinear line approaching the CEC asymptotically (Fig. 4A). A plot of CRCa versus ExCa/CEC would also produce a curvilinear line asymptotically approaching 1. The pathway from ExCa/CEC = 0 to ExCa/CEC = 1 depends only on Ca Mgor (Fig 4B). Similar conclusions apply to all other homovalent cation-exchange reactions (e.g., K+ - NH4 and K+ - Na+). 

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