Other Nonlinear Techniques

Neural network approaches have been used as an alternative to other nonlinear techniques for modeling physiological systems [Chon et al., 1998]. Several neural network control systems have utilized model-based approaches in which the neural network is used to identify a forward nonlinear system... 

Finally, other nonlinear techniques can be used to obtain spectra that are both infrared and Raman active. Infrared sum frequency generation (IRSFG) is a surface-specific nonlinear technique that relies on The coherently generated output beam has a frequency of 0)3 = a)j -I- a>2, where 0)3 is in the infrared region. The selection rules for IRSFG require that the medium be anisotropic and that the transition be both IR and Raman active. 

The autocatalytic reaction mechanism apparent at low temperatures is expected to apply to catalytic hydrogen oxidation at high pressures. In addition, the above study is the first to use STM to observe the formation of dynamic surface patterns at the mesoscopic level, which had previously been observed by other imaging techniques in surface reactions with nonlinear kinetics [57]. This study illustrates the ability of in situ STM to visualize reaction intermediates and to reveal the reaction pathway with atomic resolution. 

In one application (Maitre, 1981), the parallelism of noncoherent optical convolution methods is combined with video techniques in a totally analog application to images. Matsuoka et al (1982) have developed a hybrid optical-video-digital system based on similar principles. Pipeline processors that perform image convolution at high speed are ideal for digital applications. The inherent simplicity of the method adapts it to high-speed implementations by use of other hybrid techniques. It is even possible to devise a transversal filter that has all the desirable nonlinear properties of the method. We describe this filter in the next section. 

One of the main advantages of CARS and also of other nonlinear Raman spectroscopies is the high resolution that can be achieved in spectra of gases at low pressures. The reason for this is that the instrumental resolving power in these techniques depends only on the convoluted linewidths of the lasers used for excitation, whereas in linear Raman spectroscopy the resolution is determined by the monochromators used to disperse the observed scattered Raman light. 

In section 3.2.3, finite difference solutions were obtained for nonlinear boundary value problems. This is a straightforward and easy technique and can be used to obtain an initial guess for other sophisticated techniques. This technique is important because it forms the basis for the method of lines technique for solving linear and nonlinear partial differential equations (chapter 5 and 6). However, for stiff boundary value problems, this technique may not work and might demand prohibitively large number of node points. In addition, approximate initial guess should be provided for all the node points for stiff boundary value problems. 

Eq. (20) can be discretized in the computational domain bounded hy = o2,o ) and r] = (0,7t). The discretization can be carried out using either the finite volume or finite difference scheme, producing a system of nonlinear equations for the reduced potential in the computational domain, which can be solved employing the Newton-Ralphson method or other relaxation techniques. The boundary conditions of constant surface potentials are described by 0 = (p at the surface of sphere 1 with = o and r] = (0,7t), and (p = (p2 the surface of sphere 2 with — 02 and r] — 0,n). The boundary conditions of constant surface charge are described by... 

While this methodology has been shown to be very useful for many crystallization processes, it is subject to size dispersion and the other nonlinear effects of Fig. 4-24, and has the additional need to assure (perfect) mixing up to and including the product exit line. Use of large amounts of material is often impossible for pharmaceutical products in the early stages of development, so this technique has only limited use in our industry. 

Other nonlinear optical spectroscopies have gained much prominence in recent years. Two techniques in particular have become quite popular among surface scientists, namely, second harmonic (SHG) [55] and sum-frequency (SFG) [56] generation. The reason why both SHG and SFG can probe interfaces selectively without being overwhelmed by the signal from the bulk is that they rely on second-order processes that are electric-dipole forbidden in centrosymmetric media by breaking the bulk symmetry, the surface places the molecular species in an environment where their second-order nonlinear susceptibility, the term responsible for the absorption of SHG and SFG signals, becomes non-zero. 

The contribution of MS to identification of compounds and quantification of their concentration is complementary to other detection techniques and, despite being very practical and versatile, it remains fundamentally replaceable. However, knowledge of molecular weight is a prerequisite for techniques that rely on the synergies with stable isotopic tracers. In fact, powerful analytical methods exist to obtain important insights on cell dynamics from the ratiometric measurement of marked and not-marked species (or atoms). We cite, for example, (1) relative abundances of virtually all metabolites or proteins in two separate cultures are quantified based on the isotope dilution theory [43 5] (2) information on the mechanisms and kinetics of nonlinear chemical processes can be extracted from response tracer experiments [46 7] and (3) the labeling patterns in metabolic intermediates are used to resolve the relative rate in convergent reactions in vivo [48,49]. 

The next step in the protocol answers the question about what is the best method to estimate the reactivity ratios. Historically, because of its simplicity, linearization techniques such as the Fineman-Ross, Kelen-Tudos, and extended Kelen-Tudos methods have been used. Easily performed on a simple calculator, these techniques suffer from inaccuracies due to the linearization of the inherently nonlinear Mayo-Lewis model. Such techniques violate basic assumptions of linear regression and have been repeatedly shown to be invalid [117, 119, 126]. Nonlinear least squares (NLLS) techniques and other more advanced nonlinear techniques such as the error-in-variables-model (EVM) method have been readily available for several decades [119, 120, 126, 127]. 

For other nonlinear curve modelling techniques see D. Satyajit, Risk Management and Financial Derivatives, (New York McGraw-Hill, Inc., 1998). 

In the previous subsections we briefly introduced some nonlinear techniques of Raman spectroscopy. Besides stimulated Raman spectroscopy, Raman gain spectroscopy, inverse Raman spectroscopy, and CARS, several other special techniques such as the Raman-induced Kerr effect [361] or coherent Raman ellipsometry [362] also offer attractive alternatives to conventional Raman spectroscopy. 

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