Non-linear behaviour

In recent years there has been much activity to devise methods for multivariate calibration that take non-linearities into account. Artificial neural networks (Chapter 44) are well suited for modelling non-linear behaviour and they have been applied with success in the field of multivariate calibration [47,48]. A drawback of neural net models is that interpretation and visualization of the model is difficult. Several non-linear variants of PCR and PLS regression have been proposed. Conceptually, the simplest approach towards introducing non-linearity in the regression model is to augment the set of predictor variables (jt, X2, ) with their respective squared terms (xf,. ..) and, optionally, their possible cross-product... 

The exponential term gives rise to the highly non-linear behaviour in reactor systems which are subject to temperature changes. 

This example shows that the reactor may oscillate, either naturally according to the system parameters, or by applied controller action. Owing to the highly non-linear behaviour of the system, it is sometimes found that the net yield from the reactor may be higher under oscillatory conditions than at steady state (see simulation examples OSCIL and COOL). It should be noted also that under controlled conditions, Tset need not necessarily be set equal to the steady-state value, T, and Tset, and that the control action may be used to force the reactor to a more favourable yield condition than that simply determined by steady-state balance considerations. 

The method is better suited to dilute, rather than concentrated suspensions, since the data can be obtained in a short time (a few minutes at most) and only thin filter cakes are formed. With higher concentrations, much longer filtration times are needed and the thicker filter cakes are more likely to show compressibility effects and non-linear behaviour. 

For a TSMR sensor, the partition coefficient turns out to be proportional to the overall sensitivity of the sensor (S = f) [24], For some sensors a non-linear behaviour between sensor response and concentration is expected in particular, for those cavitands functionalized to improve hydrogen bond interactions (labelled as 1 and 3 in Figure 8). In these cases, being limited by the number of interaction sites, the characteristic is expected to be steeper at low concentrations and then to reduce the slope at higher concentrations when the interaction sites... 

SIMPLE CONSTITUTIVE EQUATIONS 1.2.1 Linear and Non-linear Behaviour... 

The data has been superimposed by dividing the relaxation function G(t) by G(t = 0), the limiting short time value, and the time has been divided by the characteristic relaxation time Tr. The first feature to notice is that the stress relaxation function overshoots and shows a peak. This is an example of non-linear behaviour. It is related to both the material and the instrumental response (Section 4.5.1). The general shape of the curves (excluding the stress overshoot) can be described using two approaches. 

A similar situation can occur when the chiral complex (containing a chiral bidentate for instance) can form dimeric species. These dimers may either be the catalyst or just an inactive resting state. Since free energies and rate constants for meso and racemic dimeric species may differ, non-linear behaviour of e.e. of ligand versus e.e. of product can result. 

There was no evidence of a second-order term in amine, nor did amine self-association account for the non-linear behaviour. Hammett p values (for variation of RNHSO2) determined for formation of the complex [S.amine] (p = 1.64) and for expulsion of the anion ( ONp) (pacyi = -1-78) are consistent with an E cB process and uncomplicated by any steric effects of bound amine in the complex. The value of Pacyi is identical with that reported previously for ElcB reaction of the same esters in 50% acetonitrile-water and much greater than for their 2-type reactions in chloroform. Consequently, an ElcB mechanism involving extensive S-O bond cleavage with the formation of a A(-sulfonylamine, ArN=S02, is supported. 

For an organic material to show electro-optical activity it must exhibit high second order non-linear behaviour. This is one of the terms in the equation that defines the polarisation (u) of an organic molecule as shown below, where E is the perturbing electric field. 

The effects of bioethanol use on NOx emissions are not consistent between different vehicles and studies. Larsen et al. [37] provided an overview of available studies, and concluded that bioethanol use can lead to either increases or decreases in NOx emissions during tests, depending on the experimental conditions. This is consistent with the non-linear behaviour of the emission-control system in petrol vehicles, where the smallest deviations from stoichiometry greatly affect NOx emissions. Leaner mixtures lead to an increase in NOx emissions, and richer mixtures lead to a decrease in NOx emissions. This erratic behaviour suggests that any direct fuel effects are masked by the ability of the fuelling system to maintain stoichiometry when changing from petrol to bioethanol blends in the different vehicles tested. 

Pokrovskii VN, Kokorin YuK (1987) The theory of oscillating birefringence of solutions of linear polymers. Dilute and concentrated systems. Polym Sci USSR 29 2385-2393 Pokrovskii VN, Kruchinin NP (1980) On the non-linear behaviour of linear polymer flow. 

In summary, from the direct observation of the vertical propagation of femtosecond-terawatt laser pulses in the atmosphere, it is demonstrated that not only the filaments, but also the whole beam, have a non-linear behaviour [18]. The overall beam is partially recollimated near to the non-linear focus. Also, filamentation can occur at 2 km altitude, which is essential for remote atmospheric applications of filamentation. 

Compared to glass, glass-ceramics are mechanically stronger, more resistant to chemical attack and have a wider range of thermal expansion coefficients. Complex non-linear thermal expansion characteristics can be achieved, resulting in very close thermal expansion matching to a variety of metals and alloys, including those with non-linear behaviour [24],... 

Comparisons of values quoted in the literature for the physical properties of liquid crystals are often of dubious validity due to differences in the methods of assessment often carried out at different absolute temperatures (e.g. 22°C or 25°C) or reduced temperatures (e.g. T -j— 10°C or 0.95 x 7V /). The use of extrapolated data from a wide variety of nematic mixtures of different composition and properties at various concentrations is also common. Unfortunately non-ideal behaviour is common for such mixtures and non-linear behaviour is not unusual, i.e. the values extrapolated to 100% are more often than not dependent on the matrix used and the concentration of the compound to be evaluated. However, although the absolute values of the data collated in Table 3.13, measured in the same way at the same reduced temperature (0.96 X r r-/), are lower than those reported for the same compounds in the literature, usually measured at 22°C the trends and relative values are very similar. 

Non-linear phenomena accompanied by periodic changes of electrochemical potential have been the subject of many research activities since Dupeyrat and Nakache [39] reported on periodic macroscopic movements of an oil/water interface and generation of electrochemical potential in 1978. These authors found such non-linear behaviour at a W/NB interface with positively charged cationic surfactants. They explained the nonlinear behaviour on the basis of formation of ion pairs between the positively charged cationic surfactants in the aqueous phase and negatively charged picrate anions dissolved in the oil phase. The ion pairs formed at a W/NB interface were assumed to be removed from the interface by a phase transfer process and oscillatory behaviour was explained in terms of the Marangoni effect. 

Later, Yoshikawa and Matsubara [40] further studied a non-linear system and proposed a mechanism for the periodic behaviour that involved the formation of inverted micelles that suddenly moved to the oil phase after the concentration of adsorbed surfactants reached a critical value. They extended the experiment to a water/oil/water three-phase system in a U-shaped glass tube that gave spontaneous and stable oscillatory behaviour over a long period [41]. Since then, various characteristics of non-linear behaviour have been investigated and several mechaiusms for the non-linear behaviour have been proposed by many research groups including ours[2,5,10,42-48] however, the mechanism at a molecular level has not been clarified yet and no consensus has been achieved. The difficulty in the explanation seems to come from not only the complexity and diversity of the systems, but also limitations of the observation methods that enable us to monitor dynamic molecular behaviour at liquid/liquid interfaces with sufficient interfacial selectivity and time resolution. In this section, the TR-QELS method has been applied to the investigation of W/NB—sodium dodecyl sulfate (SDS) two-phase system [10]. 

Because of a smaller rotation of 16° in the propeller conformation (E-conformation), the conformation dependence of the substituent effect correlations is not very serious in this system. Nevertheless, the non-linear behaviour should be similar to that in the trityl carbocation system. From comparison of the plots in Figs 8 and 15, the plot for Y = p-MeO for the benzhydryl cation should be related to the correlation for the T-conformation and that for Y = P-NO2 should be related to the P-conformer correlation. The difference in the slopes gives no clue as to the intrinsic selectivity (p) of this system. 

Perreux D., Lazuardi D., The effects of residual stress on the non-linear behaviour of composite laminates. Part I. Experimental results and residual stress assessments. Composites Science and Technology 61(2) (2001) 167-175... 

For alkyl sulphates in the absence of salt both D and Oq will be larger, leading to larger curvature effects. The observed discrepancy here however is about 400 cal mol (CH2 group), and this is too large a discrepancy to be accounted for by the simple capacitance model. At very low salt concentrations with ionic surfactants we expect highly non-linear behaviour for the repulsive electrostatic terms, so that this simple model can be expected to break down. 

Exotic oscillatory and other types of non-linear behaviour are also features of most hydrocarbon oxidations [71-74]. The next chapter will provide a detailed mechanistic description of the basis for cool-flames etc., and their relevance in various situations. It is interesting, however, to apply the classification system developed in the previous sections to the global behaviour in these systems. We start with a description of the oxidation of acetaldehyde (ethanal) and again concentrate on modern studies in flow reactors where the effects of reactant consumption (which are much more significant in closed systems for these cases than for CO) are not a feature. 

The results above suggest that it may be possible to apply fracture mechanics data to determine failure loads of more complex structures, provided that (i) the adhesives used are not too ductile, (ii) bondline thickness is known and controlled, (iii) non-linear behaviour due to adherend and interface damage is limited, and (iv) the specimens employed to determine... 

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