Mathematical algorithms

The discovery of the phenomenon that is now known as extended X-ray absorption fine structure (EXAFS) was made in the 1920s, however, it wasn t until the 1970s that two developments set the foundation for the theory and practice of EXAFS measurements. The first was the demonstration of mathematical algorithms for the analysis of EXAFS data. The second was the advent of intense synchrotron radiation of X-ray wavelengths that immensely facilitated the acquisition of these data. During the past two decades, the use of EXAFS has become firmly established as a practical and powerfiil analytical capability for structure determination. ... 

CCPS has developed guidelines for the measurement of PSM performance that provide useful ideas on measurement. Their approach for developing performance measures integrates Quality Management with a well-tested methodology for developing measures based on mathematical algorithms derived from expert assessment of system performance. The purpose was to identify well-defined, measurable, and practical indicators of performance that are updated on a schedule which makes the performance measure continuously useful. To date, measures have been developed for... 

In general, the topology of interprocessor communication reflects both the structure of the mathematical algorithms being employed and the way that the wave packet is distributed. For example, our very first implementation of parallel algorithms in a study of planar OH - - CO [47] used fast Fourier transforms (FFTs) to compute the action of 7, which also required all-to-all communication but in a topology that is very different from the simple ring-like structure shown in Fig. 5. 

While this chapter has described two examples of our efforts in bacterial strain identification, a number of other groups have contributed to this research area. It remains to be seen whether mathematical algorithms such as the modified correlation approach described herein are always more effective for strain assignments than the simple use of distinctive biomarker peaks. Nilsson reported MALDI mass spectra of Helicobacter pylori with three different matrices and solvent conditions.59 She showed that some strains of this bacteria yield rather similar mass spectra while others are quite different. Nevertheless, each strain appears to exhibit some unique peaks that might be used for distinguishing strains. 

The basic hypothesis of a QSAR model is that the activity (or effect or property) can be put in relationship with the chemical, using some parameters to describe the chemical. Thus, the three main components of the QSAR model are the activity to be modeled, the chemical information, and the way to establish a link between these two components. For this, we need some suitable ways to describe the chemical and a good mathematical algorithm. 

Optimization methods calculate one best future state as optimal result. Mathematical algorithms e.g. SIMPLEX or Branch Bound are used to solve optimization problems. Optimization problems have a basic structure with an objective function H(X) to be maximized or minimized varying the decision variable vector X with X subject to a set of defined constraints 0 leading to max(min)//(X),Xe 0 (Tekin/Sabuncuoglu 2004, p. 1067). Optimization can be classified by a set of characteristics ... 

In mathematical-algorithmic notation NIPALS can be described as follows ... 

Grammer, K., Kruck, K. Magnusson, M. 1996. The courtship dance mathematical algorithms for pattern detection in non-verbal behaviour. J. nonverb. Behav. (under revision). 

A linear series of metabolite compartments used in mathematical algorithms for modeling the kinetic behavior of metabolite and drug turnover. 

Any mathematical algorithm or technique for estimating the magnitude of f(x) for a continuous function of the variable x or for evaluating a parameter lying within an interval for which the first and last values of the interval are known. 

Clearly, an expert interpretation of teratology data cannot be reduced to a mathematical algorithm or flowchart. The teratologist uses his knowledge of the test system to make sense of the fluctuations observed and also to rule out any implausible effects. For instance, a difference in pre-implantation loss in a routine teratology study with the start of dosing on gestation day (CD) 6 is unlikely to be due to the test item. Likewise, a treatment-related reduction in resorption incidence is implausible since zero loss is within the limits of normality. 

There is however, a way conceivable to avoid these difficulties, namely the combination of GPC- and PDC-measurements performed with the same sample for which the resolution of the GPC-column is good at a possibly narrow MWD. Since the mathematical structure of the spreading functions of the GPC- and the PDC-column is the same, the parameters of Eq. (44a) (e.g. D(P), ctD(P), yD(P) and SD(P)) can then be fitted for GPC by comparing the MWDs calculated from GPC- and PDC-measurements on the same sample by the standard method shown below. Although inverted integral transforms would have to be included in such a non-linear fit, it should not be too hard to find a suitable mathematical algorithm for that iteration. However, so far no efforts have been made in this direction. 

Spath, 1991] Spath, H. (1991). Mathematical Algorithms for Linear Regression. Academic Press. 

Fig. 8.7. Different mathematical algorithms for determining the contribution of S-phase nuclei to a DNA flow histogram. Upper left and lower right from Dean (1987) upper right and lower left from Dean (1985).

Good discussions of the mathematical algorithms for cell cycle analysis can be found in Chapter 6 of Van Dilla et al., Chapter 23 of Melamed et al., and in the multiauthor book Techniques in Cell Cycle Analysis, edited by Gray JW and Darzynkiewicz Z (1987), Humana Press, Clifton, NJ. 

S-FIT An S-FIT approximation is a mathematical algorithm for guessing at which cells in a DNA-content histogram are actually in the S phase of the cell cycle. The S-FIT algorithm bases this guess on the shape of the DNA histogram in the middle region between the G0/G1 and the G2/M peaks. 

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