Linear prediction extensions

The same raw data was used in each spectrum, with the F (carbon) dimension processed with (a) no data extension, (b) one zero-fill and (c) linear prediction in place of zero-filling. 

The corresponding relation is shown in Figure 1.8. It illustrates a general feature of the elastic behavior of mbbery solids although the constituent chains obey a linear force-extension relationship (Fq. (1.1)), the network does not. This feature arises from the geometry of deformation of randomly oriented chains. Indeed, the degree of nonlinearity depends on the type of deformation imposed. In simple shear, the relationship is predicted to be a linear one with a... 

The simplest nonlinear tube model is the classical Doi-Edwards (DE) constitutive equation for linear polymers, which accounts for reptation and affine rotation of tube segments. The Doi-Edwards equation predicts thinning in both shear and extension, because it accounts for orientation of tube segments, but it is unable to predict extension thickening because it neglects the stretching of tube segments. Inclusion of tube stretch leads to the Doi-Edwards-Marrucci-... 

The discussions of DW are on p. 69 and 181-192 of Draper and Smith (third edition -the second edition contains a similar but somewhat less extensive discussion). They also include an algorithm and tables of critical values for deciding whether the correlation is statistically significant or not. You might also want to check out page 64 for the proof that the linear correlation between residuals and predicted values from the calibration is zero. 

Beside mid-IR, near-IR spectroscopy has been used to quantitate polymorphs at the bulk and dosage product level. For SC-25469 [34], two polymorphic forms were discovered (a and /3), and the /3-form was selected for use in the solid dosage form. Since the /3-form can be transformed to the a-form under pressure by enantiotropy, quantitation of the /3-form in the solid dosage formulation was necessary. Standard mixtures of both forms in the formulation matrix were prepared, and spectra were measured in the near-IR via diffuse reflectance. Utilizing a standard, near-IR multiple linear regression, statistical approach, the a- and /3-forms could be predicted to within 1% of theoretical. This extension of the diffuse reflectance IR technique shows that quantitation of polymorphic forms at the bulk and/or dosage product level can be performed. 

Subsequently Furnstahl [20] in a more extensive study pointed out that within the framework of mean field models (both non-relativistic Skyrme as well as relativistic models) there exists an almost linear empirical correlation between theoretical predictions for both 04 and its density dependence, po, and the neutron skin, All lln — Rp, in heavy nuclei. This is illustrated for 208Pb in Fig. 5 (from ref.[20] a similar correlation is found between All and po). Note that whereas the Skyrme results cover a wide range of All values the RMF predictions in general lead to AR > 0.20 fm. 

Extensive collections of pK values are available in the literature, e.g., [98-101]. It is also possible to predict pK values for a broad range of organic acids and bases using linear free energy relationships based on a systematic treatment of electronic (inductive, electrostatic, etc.) effects of substituents which modify the charge on the acidic and basic center. Quantitative treatment of these effects involves the use of the Hammett Equation which has been a real landmark in mechanistic organic chemistry. A Hammett parameter (a), defined as follows ... 

In the two classic viscometric deformations of simple shear and extension, the appropriate components of Q have very different behaviour. For small shear strains, the shear stress depends on the component Q which has the linear asymptotic form 47/15. This prefactor is the origin of tne constant v in the tube potential of Sect. 3.For large strains, however, Qxy 7 and therefore predicts strong shear-thinning. Physically this comes from the entanglement loss on re-... 

Today, well over 100 biological parameters of mammals are known to be linearly related to body weight and highly predictable on an mterspecies basis (Davidson et al. 1986, Voisin et al. 1990, Calabrese et al. 1992). The allometric equation has traditionally been used for extrapolation of experimental data concerning physiological and biochemical functions from one mammalian species to another. In addition, the allometric equation has also been used extensively as the basis for extrapolation, or scaling, of e.g., a NOAEL derived for a chemical from studies in experimental animals to an equivalent human NOAEL, i.e., a correction for differences in body size between humans and experimental animals. 

There are many examples of the linear additivity of substituent shifts. Shoolery s constants predict methylene proton shifts in disubstituted methanes for a wide range of substituents including the halogens, OB, SR, NOj, and singly, doubly, and triply bonded carbon 67). Additivity is found extensively in shifts 88). Pairwise additivity, attributed to interaction between neighboring substituents, has been demonstrated in iR 87), 2 A1 (55), Ge 41), Ti 40), and Nb 42), as well as in... 

The most relevant property of stereoregular polymers is their ability to crystallize. This fact became evident through the work of Natta and his school, as the result of the simultaneous development of new synthetic methods and of extensive stractural investigations. Previously, the presence of crystalline order had been ascertained only in a few natural polymers (cellulose, natural rubber, bal-ata, etc.) and in synthetic polymers devoid of stereogenic centers (polyethylene, polytetrafluoroethylene, polyamids, polyesters, etc.). After the pioneering work of Meyer and Mark (70), important theoretical and experimental contributions to the study of crystalline polymers were made by Bunn (159-161), who predicted the most probable chain conformation of linear polymers and determined the crystalline structure of several macromolecular compounds. 

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