Non-linear bases

it is found that certain data profiles are best described by non-linear bases. A typical example is curves from e.g. infrared spectroscopy [20] which are well described by a linear combination of Lorentzian peaks. Sometimes it is also possible to use Gaussian peaks. The Lorentzian peak function is written as 

The peak parameter representation (PPR) [21] uses only the set of non-linear basis function parameters in the multivariate modelling. The PPR method can be written as follows  

The application of wavelet transforms to the analysis of spectra and other chemical data has gained popularity in chemistry [22-30]. Examples of typical chemical signal processing application are  

There is a whole family of different wavelet methods available depending on the signal properties and the type of information that is to be extracted. However, this chapter will only focus on the fast wavelet transform (FWT) which is based on Mallat s algorithm [39.40]. It should be mentioned that the described methods to achieve parsimonious models are not dependent on one particular type of wavelet transform. Other types of wavelet transforms can be used. FWT is not always optimal for all types of problems and other techniques such as wavelet packets [41], continuous transforms [42,43] and biorthogonal transforms [37] should be considered. Some of the properties of the FWT that makes it an attractive transform are  

Choosing optimal wavelet bases. Unlike the Fourier transform there are several basis functions to select from when using wavelet transforms. This means there must be a criterion for choosing the optimal wavelet. A reasonable criterion is to use the compression ability of the analysing wavelet. This means that the optimal wavelet is defined to be the one that produces the smallest number of coefficients needed to describe the data. The following algorithm is here used  

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Raman spectroscopy has been widely used to study the composition and molecular structure of polymers [100, 101, 102, 103, 104]. Assessment of conformation, tacticity, orientation, chain bonds and crystallinity bands are quite well established. However, some difficulties have been found when analysing Raman data since the band intensities depend upon several factors, such as laser power and sample and instrument alignment, which are not dependent on the sample chemical properties. Raman spectra may show a non-linear base line to fluorescence (or incandescence in near infrared excited Raman spectra). Fluorescence is a strong light emission, which interferes with or totally swaps the weak Raman signal. It is therefore necessary to remove the effects of these variables. Several methods and mathematical artefacts have been used in order to remove the effects of fluorescence on the spectra [105, 106, 107]. 

Low and High frequency can be restored by use of a deconvolution algorithm that enhances the resolution. We operate an improvement of the spectral bandwidth by Papoulis deconvolution based essentially on a non-linear adaptive extrapolation of the Fourier domain. 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

For each active node in the current mesh the corresponding location array is searched to find inside which element the foot of the trajectory currently passing through that node is located. This search is based on the. solution of the following set of non-linear algebraic equations... 

Thiirane 1,1-dioxides extrude sulfur dioxide readily (70S393) at temperatures usually in the range 50-100 °C, although some, such as c/s-2,3-diphenylthiirane 1,1-dioxide or 2-p-nitrophenylthiirane 1,1-dioxide, lose sulfur dioxide at room temperature. The extrusion is usually stereospeciflc (Scheme 10) and a concerted, non-linear chelotropic expulsion of sulfur dioxide or a singlet diradical mechanism in which loss of sulfur dioxide occurs faster than bond rotation may be involved. The latter mechanism is likely for episulfones with substituents which can stabilize the intermediate diradical. The Ramberg-Backlund reaction (B-77MI50600) in which a-halosulfones are converted to alkenes in the presence of base, involves formation of an episulfone from which sulfur dioxide is removed either thermally or by base (Scheme 11). A similar conversion of a,a -dihalosulfones to alkenes is effected by triphenylphosphine. Thermolysis of a-thiolactone (5) results in loss of carbon monoxide rather than sulfur (Scheme 12). 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

In an alternative approaeh, Jezequel (2001) proposes the eoneept of the sealeable reaetor based on the observation that in highly non-linear systems sueh as preeipitation, seale-up is better assured when all the eritieal parts of the proeess are exaetly at the same seales. Thus, the eritieal parts are sealed by replieation, whilst the main holding vessel in whieh slow proeesses take plaee is sealed in volume (Figure 8.9). 

The linear ramp instrument uses a linear time-base to determine the time taken for an internally generated voltage to reach the unknown voltage, V. The limitations are due to small non-linearity in the ramp, instability of the electronic components and a lack of noise rejection. 

Tantalum and niobium are added, in the form of carbides, to cemented carbide compositions used in the production of cutting tools. Pure oxides are widely used in the optical industiy as additives and deposits, and in organic synthesis processes as catalysts and promoters [12, 13]. Binary and more complex oxide compounds based on tantalum and niobium form a huge family of ferroelectric materials that have high Curie temperatures, high dielectric permittivity, and piezoelectric, pyroelectric and non-linear optical properties [14-17]. Compounds of this class are used in the production of energy transformers, quantum electronics, piezoelectrics, acoustics, and so on. Two of... 

We employed a comparative Z-scan procedure, wherein we perform a reduced-aperture Z-scan on CS2 immediately followed by reduced and open-aperture Z-scans on the polymer solution at a particular wavelength. The peak on-axis intensity, /0, is then calculated from the CS2 peak to valley transmission change using previously measured values for its non-linear axis of refraction. Based on the work of Sheik-Bahae et al the equation for /0is given by 37]... 

Subsequently it was shown that the P-Pd-P angles were essentially the same as in the corresponding chloride complexes (section 3.8.3) as a result, as the P—Pd—P angle increases, concomitant upon the increase in the length of the methylene chain, steric effects enforce N-bonded thiocyanate, which is less sterically demanding that the non-linear Pd-SCN linkage (favoured on HSAB considerations since Pd2+ is a soft acid and sulphur is a soft base). 

Instead of using repeated solution of a suitable eigenvalue equation to optimize the orbitals, as in conventional forms of SCF theory, we have found it more convenient to optimize by a gradient method based on direct evaluation of the ener functional (4), ortho normalization being restored after every parameter variation. Although many iterations are required, the energy evaluation is extremely rapid, the process is very stable, and any constraints on the parameters (e.g. due to spatial symmetry or choice of some type of localization) are very easily imposed. It is also a simple matter to optimize with respect to non-linear parameters such as orbital exponents. 

Purcell and coworkers obtained good correlations between the IR and H NMR methods by the use of 1,1,1,3,3,3-hexafluoropropanol and phenol as acids . Taddei and colleagues have reported that with CHCI3 as the donor, a free energy relationship is established between the association constants and the H NMR chemical shift values of CHCI3 in the presence of Lewis bases. However, a non-linear relationship has been observed for several other Lewis bases and CHCI3 by other authors . 

Additionally, Breiman et al. [23] developed a methodology known as classification and regression trees (CART), in which the data set is split repeatedly and a binary tree is grown. The way the tree is built, leads to the selection of boundaries parallel to certain variable axes. With highly correlated data, this is not necessarily the best solution and non-linear methods or methods based on latent variables have been proposed to perform the splitting. A combination between PLS (as a feature reduction method — see Sections 33.2.8 and 33.3) and CART was described by... 

Basically two search procedures for non-linear parameter estimation applications apply. (Nash and Walker-Smith, 1987). The first of these is derived from Newton s gradient method and numerous improvements on this method have been developed. The second method uses direct search techniques, one of which, the Nelder-Mead search algorithm, is derived from a simplex-like approach. Many of these methods are part of important mathematical computer-based program packages (e.g., IMSL, BMDP, MATLAB) or are available through other important mathematical program packages (e.g., IMSL). 

The linearisation of the non-linear component and energy balance equations, based on the use of Taylor s expansion theorem, leads to two, simultaneous, first-order, linear differential equations with constant coefficients of the form... 

Van Westrenen et al. (2001a) present a model of lanthanide and Sc partitioning between the garnet X-site and melt. The model is a variant of the lattice strain model of clinopyroxene-melt partitioning of Wood and Blundy (1997), and is based on 160 experimental garnet-melt pairs in the pressure-temperature range 2.5-7.5 GPa and 1450-1930°C. The model includes composition-sensitive expressions for and accounts for the non-linear variation in with composition, as follows ... 

Reliable chronologies should be based on multiple °Th age determinations at a sampling interval that is appropriate for the spatial resolution of geochemical analysis along the axis of growth and accounts for the possibility of non-linear or discontinuous growth. 

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