Early Constrained Methods

It has been noted that deconvolution methods, most of which were linear, had a propensity to produce solutions that did not make good physical sense. Prominent examples were found when negative values were obtained for light intensity or particle flux. As noted previously, the need to eliminate these negative components was generally accepted. Accordingly, Gold (1964) developed a method of iteration similar to Van Cittert s but used multiplicative corrections instead of additive ones. 

A ratio containing this quantity and the data may be used to construct a new estimate of the true spectrum, 

As with Van Cittert s method, we may taked(0) = i. We see that the successive estimates dik) cannot be negative, provided that s and i are everywhere positive. The reader should note that this assumption may be violated in a base-line region. Here the data i probably contain negative values arising from noise. A means of dealing with this problem is needed. 

Gold s method has been used by a number of workers, including Siska (1973), who applied it to molecular-beam scattering data, MacNeil and Dixon (1977), who applied it to photoelectron spectra, and Jones et al. (1967), who restored infrared spectra of condensed-phase samples. The author is unaware of any experimental results with this method, however, that illustrate the full potential achievable by constrained methods to be described later in this chapter. In the work of Jones et al., the resulting resolution is probably limited by the inherent breadth of spectral lines observed with condensed-phase samples. 

Here o and i are the object and image vectors, respectively, and s is the spread-function matrix as defined in Chapter 3. Gold was able to show that proper convergence is assured if the following conditions hold  

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Knowing that the better nonlinear constrained methods are now available, why have researchers generally been reluctant to accept them Perhaps the linear approach has an attraction that is not related to performance. Early in a technical career the scientist-engineer is indoctrinated with the principles of linear superposition and analysis. Indeed, a rather large body of knowledge is based on linear methods. The trap that the linear methods lay for us is the existence of a beautiful and complete formalism developed over the years. Why complicate it by requiring the solution to be physically possible ... 

Perhaps the benefits of physical-realizability constraints, particularly ordinate bounds such as positivity, have not been sufficiently recognized. Surely everyone agrees in principle that such constraints are desirable. Even the early literature on this subject frequently mentions their potential advantages. For one reason or another, however, the earliest nonlinear constrained methods did not fully reveal the inherent power of constraints. 

Powder flow is most frequently thought of as relevant to formulation development, and there are numerous references attempting to correlate any one of a number of measures of powder flow to the manufacturing properties of a formulation [34—40]. In particular, the importance of physical properties in affecting powder flow has been well documented. Research into the effect of the mechanical properties on powder flow has, however, been very limited. It is, of course, important to be able to determine and quantitate the powder flow properties of formulations. It is of equal importance, however, to determine the powder flow characteristics of bulk drug early in the development process (preformulation phase). Often, the preformulation or formulation scientist is constrained by time, materials, and manpower. Yet certainly the preformulation studies carried out should be meaningful. Well-defined experimental methods and procedures should be used the information generated should be reproducible and permit useful predictions to be made. 

Whereas preparation of a-amino acid derivatives by asymmetric allylation of an acyclic iminoglycinate gave a modest enantioselectivity (62% ee) in an early investigation [189], the use of conformationally constrained nucleophiles in an analogous alkylation resulted in high selectivities (Scheme 8E.43) [190], With 2-cyclohexenyl acetate, the alkylation of azlactones occurred with good diastereomeric ratios as well as excellent enantioselectivities. This method provides very facile access to a variety of a-alkylamino acids, which are difficult to synthesize by other methods. When a series of azlactones were alkylated with a prochiral gem-diacetate, excellent enantioselectivities were uniformly obtained for both the major and minor diastereom-ers (Eq. 8E.20 and Table 8E.12). 

Fig. 32. PAAS-normalized lanthanide patterns for early Proterozoic uranium deposits from the Pine Creek Geosyneline, Australia (data are from McLennan and Taylor 1979). Compared to the unmineralized host sedimentary roek, these deposits are extremely enriched in heavy lanthanides and depleted in light lanthanides. The abundances and amount of fractionation is related to the U content, suggesting the lanthanide mobility and U mineralization are directly related, thus helping to constrain the origin of the deposit. The fractionation of Sm and Nd during ore-formation allows dating by the Sm-Nd method.

Despite the obvious success of munerical methods for nonlinear mathematical programming, their weaknesses were discovered early on. Among them it should be highlighted the main one, namely, the absence of the physicochemical visualization. To some extent it relates also to Bellmann s dynamic programming method. Naturally, incomplete information about the nature of the studied process on a way to optimal result constrains strongly the creative capabilities of a researcher. In particular, identification of the most active control parameters from a variety of the candidates is complicated, thus also complicating the solution of the defined problem. 

Newton and Leibnitz. The foundations of calculus of variations were laid by Bernoulli, Euler, Lagrange and Weierstrass. The optimization of constrained problems, which involves the addition of unknown multipliers, became known by the name of its inventor Lagrange. Cauchy made the first application of the steepest descent method to solve unconstrained minimization problems. In spite of these early contributions, very little progress was made until the middle of the 20th century, when high-speed digital computers made the implementation of the optimization procedures possible and stimulated further research in new methods. 

Although the criteria of Talarico et al. are vaUd for every software system a number of aspiects are especially different for embedded systems. For example, the forth requirement support for a cost-effective method to formally spjeofy system constrains is difficult for embedded systems, since the exact nature of all constraints is often not yet known in the early stages of development. 

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