Relationships nonlinear

Nonlinear relationships between biological activities and lipophilicity are very common. While biological activity values most often linearly increase with increasing lipophilicity [182], such an increase is no longer obtained if a certain range of lipophilicity is surpassed biological activities remain constant or decrease more or less rapidly with further increase of lipophilicity [7, 19]. Many reviews deal with nonlinear lipophilicity-activity relationships [19, 175, 178, 345, 433]. 

Different explanations have been given for this effect, the kinetics of drug transport in biological systems and the distribution of drugs in different compartments of a biological system being the most common. In addition, 

Hansch formulated a parabolic model (eq. 7, chapter 1.1) [15, 17—19] for the mathematical description of nonlinear relationships. He was aware that the sides of a parabola are always more or less curved, while in most cases at least the left side of the structure-activity relationship (i.e. the lipophilicity dependence of the more hydrophilic analogs) is strictly linear equations including a third-order lipophilicity term did not produce much improvement [19]. A computer simulation of the transport of drugs in a biological system, using hypothetical rate constants, 

For practical reasons and to avoid too many parameters, it is better to combine both equations to eq. 84. The term log [P PJ in eq. 84 is zero for all log P values being smaller than log P while it is the difference (log P — log PJ for all log P values being larger than log P [175, 345]. 

The Franke model is a definite improvement as compared to the parabolic model. In many practical cases it gives a much better fit, e.g. eqs. 85 and 86 for the spasmolytic activities of mandelic acid esters (Table 14 RA = relative biological activity values, based on cyclandelate (15), RA = 100%) [345]. 

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Figure 26-65 illustrates that Eq. (26-90) provides a linear approximation to the nonlinear relationship between two-phase specific volume and reciprocal pressure (v vs. P or vs. T ). For single components, me initial slope of the curve is found using me Clapeyron equation to give ... 

At low strains there is an elastic region whereas at high strains there is a nonlinear relationship between stress and strain and there is a permanent element to the strain. In the absence of any specific information for a particular plastic, design strains should normally be limited to 1%. Lower values ( 0.5%) are recommended for the more brittle thermoplastics such as acrylic, polystyrene and values of 0.2-0.3% should be used for thermosets. 

The rate of temperature drop of a fluid as it flows along the length of a heat exchanger is not constant. In order to take account of this nonlinear relationship, the logarithmic mean temperature difference (EMTD) is used. If the inlet and outlet temperatures do not differ widely, an arithmetic mean can be used, because the relationship is considered to be linear. 

Hyperbola (hyperbolic), a set of functions defining nonlinear relationships between abscissae and ordinates. This term is used loosely to describe nonlinear relationships between the initial interaction of molecules and receptors and the observed response (i.e., stimulus-response cascades of cells). 

The final stages of the methodology consist of eliminating some of the less Informative tests and adding other tests to provide data needed to quantify nonlinear relationships The resulting experimental design then consists of a mix of conditional main effects and conditional Interactions It represents a compromise among scientific, statistical, and economic constraints ... 

An important issue that influences estuarine behavior is the determination of the riverine component. Variations in the riverine endmember may occur over timescales that are short compared to the residence times of water in the estuary, which result in nonlinear relationships between salinity and trace elements across the estuary as different riverine compositions progress through the estuary (Loder and Reichard 1981 Officer... 

Chemical reaction rates may show large variations from reaction to reaction, and also with changes of temperature. It is often found that one or the other of the steps involved in the overall process offers the major resistance to its occurrence. Such a slow step controls the rate of the process. As a simplification such a rate-controlling step can be considered alone. In an alternative procedure the nonlinear relationship between rate and concentration is approximated to a linear relationship. To do this the nonlinear rate is expanded in the form of a Taylor s series and only the linear terms are retained. 

When the output vector (measured variables) are related to the state variables (and possibly to the parameters) through a nonlinear relationship of the form y(t) = h(x(t),k), we need to make some additional minor modifications. The sensitivity of the output vector to the parameters can be obtained by performing the implicit differentiation to yield ... 

A strong nonlinearity relationship was observed between ee % of chiral modifier 46 and the adduct 50. 

Methods based on nonlinear projection exploit the nonlinear relationship between the inputs by projecting them on a nonlinear hypersurface resulting in latent variables that are nonlinear functions of the inputs, as shown in Figs. 6b and 6c. If the inputs are projected on a localized hypersurface such as a hypersphere or hyperellipse, then the basis functions are local, depicted in Fig. 6c. Otherwise, the basis functions are nonlocal, as shown in Fig. 6b. 

By design, ANNs are inherently flexible (can map nonlinear relationships). They produce models well suited for classification of diverse bacteria. Examples of pattern analysis using ANNs for biochemical analysis by PyMS can be traced back to the early 1990s.4fM7 In order to better demonstrate the power of neural network analysis for pathogen ID, a brief background of artificial neural network principles is provided. In particular, backpropagation artificial neural network (backprop ANN) principles are discussed, since that is the most commonly used type of ANN. 

The popular radial basis function nets (RBF nets) model nonlinear relationships by linear combinations of basis functions (Zell [1994] Jagemann [1998] Zupan and Gasteiger [1993]). Functions are called to be radial when their values, starting from a central point, monotonously ascend or descend such as the Cauchy function or the modified Gauss function at Eq. (6.125) ... 

Neural networks are applied in analytical chemistry in many and diverse ways. Used in calibration, ANNs have especially advantages in case of nonlinear relationships, multicomponent systems and single component analysis in case of various disturbances. 

Chergui K., Suaud-Chagny M. F., Gonon F. (1994). Nonlinear relationship between impulse flow, dopamine release and dopamine elimination in the rat brain in vivo. Neuroscience 62(3), 641-5. 

The proposed dinuclear transition-state model (1) has been supported by the observation of nonlinear relationship between enantiomeric excess (ee) of the epoxide and ee of DAT.33 The use of simple diol instead of tartrate vitiates stereoselectivity of the reaction.34,35 The ester group of DAT is indispensable for the construction of the desired catalyst. It is noteworthy that 1,2-di(o-methoxy-phenyl)ethylenediol is an efficient chiral auxiliary for titanium-mediated epoxidation, while 1,2-diphenylethylenediol is a poor one.36... 

Now let us consider a two-wavelength model for one of the components in a solution containing two components in a nonabsorbing solvent (hypothetical case, NOT water in the NIR ). The effect of nonlinearity in the relationship of the two components to their absorbances will have different effects. If the component being calibrated for has a nonlinear relationship, that will show up in the plot of the predicted versus actual values, as a more-or-less obvious curvature in the plot, somewhat as we showed as Figure 68-lb in our Chapter 68 [1],... 

A nonlinear relationship in the other component, however, will not show up that way. Let us try to draw a word picture to describe what we are trying to say here (the way we draw, this is by far the easier way) since we could imagine this being plotted in three dimensions, the nonlinear relation will be in the depth dimension, and will be projected on the plane of the predicted-versus-actual plot of the component being calibrated for. In this projection, the nonlinearity will show up as an extra error superimposed on the data, and will be in addition to whatever random error exists in the known values of the composition. Unless the concentrations of the other component are known, there is no way to separate the effects of the nonlinearity from the random error, however. While we cannot actually draw this picture, graphical illustration of these effects have been previously published [8],... 

Liquid scintillation counters are highly efficient for low CL intensities and consist of two photon-counting channels provided with a variable discriminator. The sample is placed between these two detectors to ensure a high optical efficiency. The discriminator is adjusted to allow photon impacts to be transmitted and small background noise pulses to be rejected. As disadvantages they suffer from saturation errors and provide nonlinear relationships between the CL intensity and the total counts. 

Thus the weights and abscissas for the average NDF cannot be found to be averaging those for the NDF in each environment. Due to the nonlinear relationship between the moments and weights and abscissas, this result is not surprising.5 However, it does illustrate that (wm) and (lm) are not the relevant quantities needed to reconstruct (mk). 

When chloroform was added, the equilibrium response of the sensor progressively decreased. This is probably related to the combination of the nonlinear behavior of the effective refractive index of the coupled cladding mode on the overlay refractive index with the nonlinear relationship between adsorbed mass of... 

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