The Importance of Linearity

One point that needs to be mentioned is that the analysis described above assumes that the sample s response to the modulation can be approximated as linear. Clearly, the processes such as those that follow Arrhenius kinetics or related kinetics of a glass transition are not linear with temperature. However, over a small temperature interval they can be approximated as linear. Where this cannot be said to be true, the above analysis fails because it assumes a linear response. 

Where a multiplexed sine wave or saw-tooth modulation is used the deconvolution procedure can be used to extract the response at a series of frequencies [4,10,19,20]. However, current commercial products restrict themselves to using the first component of the Fourier series, which is then, with the assumption of linearity, equivalent to using a single sinusoidal modulation. It is true that looking at the whole Fourier series, rather than just the first component, offers scope for increasing the amount of information that can be obtained from an MTDSC experiment. This applies even to single sinusoidal modulations (because non-linearities produce harmonics) as well as multiple simultaneous sine waves or saw-tooth modulations. This will be considered in greater detail below in the section on advanced MTDSC. 

---

The relevance of size-related properties of hERG-blocking molecules was also detected in a 2D QSAR model developed by Coi et al. [22] after the analysis of 82 compounds through the CODESSA method. These authors developed two multiparameter models with strong predictive properties, from which, besides the involvement of hydrophobic features, the importance of linearity as opposed to globularity of the hERG blockers emerged. 

Matlab recognises the importance of linear regression calculations and introduced a very elegant and useful notation the / forward- and backslash operators, see p.l 17-118. 

The importance of linear chromatography comes from the fact that almost all analytical applications of chromatography are carried out xmder such experimental conditions that the sample size is small, the mobile phase concentrations low, and thus, the equilibrixim isotherm linear. The development in the late 1960s and early 1970s of highly sensitive, on-line detectors, with detection limits in the low ppb range or lower, permits the use of very small samples in most analyses. In such cases the concentrations of the sample components are very low, the equilibrium isotherms are practically linear, the band profiles are symmetrical (phenomena other than nonlinear equilibrium behavior may take place see Section 6.6), and the bands of the different sample components are independent of each other. Qualitative and quantitative analyses are based on this linear model. We must note, however, that the assumption of a linear isotherm is nearly always approximate. It may often be a reasonable approximation, but the cases in which the isotherm is truly linear remain exceptional. Most often, when the sample size is small, the effects of a nonlinear isotherm (e.g., the dependence of the retention time on the sample size, the peak asymmetry) are only smaller than what the precision of the experiments permits us to detect, or simply smaller than what we are ready to tolerate in order to benefit from entertaining a simple model. 

Figure 4.31 The importance of linear diffusion on the kinetics of cleavage by EcoRV. EcoRV recognizes and cleaves GAT ATC sites on DNA. The overall rate of cleavage was determined for PCR-amplified DNA sequences of different length, but each containing one restriction site, (a) Model for association and one-dimensional diffusion, (b) Experimental data for relative rate of cleavage for long DNA vs short DNA (26 bp) [133].

Now that you realize the importance of linear models in describing engineering atuatiotis, let us consider some of the basic characteristics of a linear mockL As you know, the basic ferm of a line equation is given by... 

This method has gained importance in recent time in view of the importance of linear acenes in molecular electronics applications. The following synthesis of substituted heptacene demonstrates the utility of this methodology (Scheme 16.28) [31]. 

In the first section of this chapter, we emphasized the importance of linear scaling with the number of electrons in the system when developing approximate descriptions of many-body systems. We showed that both the exact solutions of the many-electron... 

The linear viscoelasticity is a well established science and textbooks are available for understanding theories and practices. Therefore, no review of the subject will be made here. However, it is appropriate to make comments on the importance of linear... 

VR, the inputs correspond to the value of the various parameters and the network is 1 to reproduce the experimentally determined activities. Once trained, the activity of mown compound can be predicted by presenting the network with the relevant eter values. Some encouraging results have been reported using neural networks, have also been applied to a wide range of problems such as predicting the secondary ire of proteins and interpreting NMR spectra. One of their main advantages is an to incorporate non-linearity into the model. However, they do present some problems Hack et al. 1994] for example, if there are too few data values then the network may memorise the data and have no predictive capability. Moreover, it is difficult to the importance of the individual terms, and the networks can require a considerable 1 train. 

Nimura, Y. Carr, M. R. Reduction of the Relative Error in the Standard Additions Method, Analyst 1990, 115, 1589-1595. The following paper discusses the importance of weighting experimental data when using linear regression Karolczak, M. To Weight or Not to Weight An Analyst s Dilemma, Curr. Separations 1995, 13, 98-104. 

The importance of unsaturation is illustrated by the fact that 2,4-nonadienoic acid [21643-39-0] forms a Hquid crystal phase, whereas the aHphatic carboxyHc acids do not. The two double bonds enhance the polarizabiHty of the molecule and bring iatermolecular attractions to a level that is suitable for mesophase formation. The overall linearity of the molecule must not be sacrificed ia poteatial Hquid crystal candidates. For example, whereas /n j - -aIkoxyciaaamic acids (5) are mesomorphic, the cis isomers (6) are not, a reflection of the greater anisotropy of the trans isomer. 

Titanium Trichloride. Titanium trichloride [7705-07-9] exists in four different soHd polymorphs that have been much studied because of the importance of TiCl as a catalyst for the stereospecific polymerization of olefins (120,124). The a-, y-, and 5-forms are all violet and have close-packed layers of chlorines. The titaniums occupy the octahedral interstices between the layers. The three forms differ in the arrangement of the titaniums among the available octahedral sites. In a-TiCl, the chlorine sheets are hexagonaHy close-packed in y-TiCl, they are cubic close-packed. The brown P-form does not have a layer stmcture but, instead, consists of linear strands of titaniums, where each titanium is coordinated by three chlorines that act as a bridge to the next Ti The stmctural parameters are as follows ... 

Reduction of oxygen is one of the predominant cathodic reactions contributing to corrosion. Awareness of the importance of the role of oxygen was developed in the 1920s (19). In classical drop experiments, the corrosion of iron or steel by drops of electrolytes was shown to depend on electrochemical action between the central relatively unaerated area, which becomes anodic and suffers attack, and the peripheral aerated portion, which becomes cathodic and remains unattacked. In 1945 the linear relationship between rate of iron corrosion and oxygen pressure from 0—2.5 MPa (0—25 atm) was shown (20). 

The series converge for all x. Much of the importance of Bessel s equation and Bessel functions lies in the fact that the solutions of numerous linear differential equations can be expressed in terms of them. 

The importance of inherent flaws as sites of weakness for the nucleation of internal fracture seems almost intuitive. There is no need to dwell on theories of the strength of solids to recognize that material tensile strengths are orders of magnitude below theoretical limits. The Griffith theory of fracture in brittle material (Griflfith, 1920) is now a well-accepted part of linear-elastic fracture mechanics, and these concepts are readily extended to other material response laws. 

From the above comments it will be expected that terephthalic acid would be an important intermediate in the production of linear crystallisable polyesters, and so it has proved. 

As shown in the introductory section to this chapter, polyesters may be produced in many ways, one of which is the method of ester exchange. This has been of importance in the manufacture of linear polyesters for a number of reasons which include the following ... 

The impact of the value of (k 2) is obviously linear, but controls the overall analysis time. It emphasizes the importance of carefully selecting the phase system not only to make the value of (a) as large as possible but also to keep (k 2) to a minimum. 

Linear alcohols (C12-C26) are important chemicals for producing various compounds such as plasticizers, detergents, and solvents. The production of linear alcohols by the hydroformylation (Oxo reaction) of alpha olefins followed by hydrogenation is discussed in Chapter 5. They are also produced by the oligomerization of ethylene using aluminum alkyls (Ziegler catalysts). 

Skold and Larson" in studies of the corrosion of steel and cast iron in natural water found that a linear relationship existed between potential and the applied anodic and cathodic current densities, providing the values of the latter were low. However, the recognition of the importance of these observations is due to Stern and his co-workerswho used the term linear polarisation to describe the linearity of the rj — i curve in the region of E o , the corrosion potential. The slope of this linear curve, AE — AJ or Af - A/, is termed the polarisation resistance, / p, since it has dimensions of ohms, and this term is synonymous with linear polarisation in... 

The importance of the method in corrosion testing and research has stimulated other work, and since Stern s papers appeared there have been a number of publications many of which question the validity of the concept of linear polarisation. The derivation of linearity polarisation is based on an approximation involving the difference of two exponential terms, and a number of papers have appeared that have attempted to define the range of validity of polarisation resistance measurements. Barnartt" derived an analytical expression for the deviations from linearity and concluded that it varied widely between different systems. Leroy", using mathematical and graphical methods, concluded that linearity was sufficient for the technique to be valid in many practical corrosion systems. Most authors emphasise the importance of making polarisation resistance measurements at both positive and negative overpotentials. 

We have shown that the number of linearly independent invariants of degree k under the permutation group H is equal to the coefficient of in the Maclaurin expansion of (1.27). This represents an important special case of a proposition by Th. Molien. ... 

---