INDEX mathematical methods

Use mathematical methods, such as Hit quality index (HQI) value (or similarity value vs. library reference sample). An example list of such HQI methods includes... 

Several multivariable controllers have been proposed during the last few decades. The optimal control research of the 1960s used variational methods to produce multivariable controllers that rninirnized some quadratic performance index. The method is called linear quadratic (LQ). The mathematics are elegant but very few chemical engmeering industrial applications grew out of this work. Our systems are too high-order and nonlinear for successful application of LQ methods. 

Examples of mathematical methods include nominal range sensitivity analysis (Cullen Frey, 1999) and differential sensitivity analysis (Hwang et al., 1997 Isukapalli et al., 2000). Examples of statistical sensitivity analysis methods include sample (Pearson) and rank (Spearman) correlation analysis (Edwards, 1976), sample and rank regression analysis (Iman Conover, 1979), analysis of variance (Neter et al., 1996), classification and regression tree (Breiman et al., 1984), response surface method (Khuri Cornell, 1987), Fourier amplitude sensitivity test (FAST) (Saltelli et al., 2000), mutual information index (Jelinek, 1970) and Sobol s indices (Sobol, 1993). Examples of graphical sensitivity analysis methods include scatter plots (Kleijnen Helton, 1999) and conditional sensitivity analysis (Frey et al., 2003). Further discussion of these methods is provided in Frey Patil (2002) and Frey et al. (2003, 2004). 

Unlike direct reference assays, indirect index methods use two separate tests to estimate the free hormone concentration a total serum T4 (or T3) measurement and an assessment of either the serum TBG concentration or the fraction of T4 (or T3) that is free m serum the latter is traditionally derived using equilibrium tracer dialysis or T uptake methods. Results of these tests are then combined mathematically to give estimates of the free hormone concentration. Indirect index-based methods have essentially fallen out of favor in deference to the direct immunoassay methods. 

Morgan Algorithm is a mathematical method for canonical (unique) numbering of atoms in a molecule based on iterative indexing of atoms according to the number of their attached bonds. 

As outlined in the introducing chapter, the minimizers and the saddle points of index one of the energy functional are corner-stones of most reaction theories in chemistry. Thus the applicability of these theories strongly depends on the availability of mathematical methods, which compute such points in an effective manner. Therefore the current chapter is engaged in the presentation of methods, which enable to compute minima and/or saddles of a PES. 

Correlation methods discussed include basic mathematical and numerical techniques, and approaches based on reference substances, empirical equations, nomographs, group contributions, linear solvation energy relationships, molecular connectivity indexes, and graph theory. Chemical data correlation foundations in classical, molecular, and statistical thermodynamics are introduced. 

In producing an image of molecules from crystallographic data, the computer simulates the action of a lens, computing the electron density within the unit cell from the list of indexed intensities obtained by the methods described in Chapter 4. In this chapter, I will discuss the mathematical relationships between the crystallographic data and the electron density. 

The basic ideas that are necessary for the first program stage are explained in Sections II, III, and IV. In Section II, we formulate the problem of how to analyze a system that has a gap in characteristic time scales. Our method is to use perturbation theory with respect to a parameter that is the ratio between a long time scale and a short time scale, which is a version of singular perturbation theory. The reason will be explained in Section II. In Section III, the concept of NHIMs is introduced in the context of singular perturbation theory. We will give an intuitive description of NHIMs and explain how the description is implemented, leaving the precise formulation of the NHIM concept to the literature in mathematics. In Section IV, we will show how Lie perturbation theory can be used to transform the system into the Fenichel normal form locally near a NHIM with a saddle with index 1. Our explanation is brief, since a detailed exposition has already been published [2]. 

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