Methods, based on matrix

Thus, these methods are suitable for problems with a very large number of parameters. They are essential in circumstances when methods based on matrix factorization are not viable because the relevant matrix is too large or too dense (Gill etal. 1981). 

An analytical method based on matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF MS) was applied to provide information on the structure of copolymer 29, for example, repeat units and end groups <2002ANC6252>. 

L. Thylen, and C. M. Lee, Beam propagation method based on matrix diagonalisation, Jowrna/ of the Optical Society of America A 9, 142-146 (1992). 

Many enzyme reactions have more than one intermediate for which King and Altman (1956) devised a method, based on matrix algebra, by establishing the rate equation of a given enzymic reaction simply by inspecting all complexes and the reactions between them. 

In some doublet reactions the same products may be obtained from the same reactants but in different ways. Such reactions were referred to as olistomeric and the structural conditions of their appearance were determined on the basis of the multiplet theory (348). Thus, theoretically two cases of bond fission can take place in esterification, HO—COR + H—OR and H—OCOR HO—R. The tracer method shows that the first case is realized as a rule. The half-doublet scheme expresses such experiments. The doublet indexes for the reactions in solutions disintegrate into such schemes see Balandin (37). A complete system of doublet reactions for C, H, N, 0, S, and Cl (without their isotopes) has been obtained by the author (345). It is much more detailed than Table VII and amounts to 1500 types. It was obtained by exhaustive variation of atoms and bonds in the index by means of a specially developed method based on matrix algebra and combinatorial analysis. The significance of the obtained classification for organic catalysis is similar to that of the complete system of forms in crystallography, which was derived from the groups of symmetry by Fedorov and covers all possible forms (349). 

Segmentation method based on the analysis by Co-Occurrence Matrix is developed. We try to increase the quality of the obtained results by means of the application of two dimensional (2D) processing. We use Co-Occurrence Matrix for ultrasonic image segmentation. This tool, introduced by Haralick (1), was selected for the present study as several general considerations were favourable ... 

The problem is then reduced to the representation of the time-evolution operator [104,105]. For example, the Lanczos algorithm could be used to generate the eigenvalues of H, which can be used to set up the representation of the exponentiated operator. Again, the methods are based on matrix-vector operations, but now much larger steps are possible. 

Yarkoni [108] developed a computational method based on a perturbative approach [109,110], He showed that in the near vicinity of a conical intersection, the Hamiltonian operator may be written as the sum a nonperturbed Hamiltonian Hq and a linear perturbative temr. The expansion is made around a nuclear configuration Q, at which an intersection between two electronic wave functions takes place. The task is to find out under what conditions there can be a crossing at a neighboring nuclear configuration Qy. The diagonal Hamiltonian matrix elements at Qy may be written as... 

Gravimetric methods based on precipitation or volatilization reactions require that the analyte, or some other species in the sample, participate in a chemical reaction producing a change in physical state. For example, in direct precipitation gravimetry, a soluble analyte is converted to an insoluble form that precipitates from solution. In some situations, however, the analyte is already present in a form that may be readily separated from its liquid, gas, or solid matrix. When such a separation is possible, the analyte s mass can be directly determined with an appropriate balance. In this section the application of particulate gravimetry is briefly considered. 

IR absorption spectra of oxypentafluoroniobates are discussed in several publications [115, 157, 167, 185, 186], but only Surandra et al. [187] performed a complete assignment of the spectra. Force constants were defined in the modified Urey-Bradley field using Wilson s FG matrix method. Based on data by Gorbunova et al. [188], the point group of the NbOF52 ion was defined as C4V. Fifteen normal modes are identified for this group, as follows ... 

It may be eoneluded that a method based on the K-matrix teehnique may be conveniently adapted to ealeulate the eontinuum properties using variational basis funetions that are aeeurate only inside the molecular region . This means that the calculations may be carried out upon GTO bases, which allow the extension of the proposed method to moleeular systems, as aheady eheeked for H2 (13). 

A final special case may occur during the validation of common moiety methods. Based on the normal set of recovery experiments (two control samples, five samples fortified at the LOQ and five samples fortified at 10 times the LOQ), in total 12 samples have to be analyzed per matrix and analyte. A typical intention of common moiety methods is their suitability for the parallel determination of residues of the parent compound and a broad spectrum of metabolites. In the common moiety method discussed above for residues of spiroxamine, validation experiments were performed with four compounds. This results in at least 48 experiments per matrix. Assuming a normal... 

Our analysis is based on solution of the quantum Liouville equation in occupation space. We use a combination of time-dependent and time-independent analytical approaches to gain qualitative insight into the effect of a dissipative environment on the information content of 8(E), complemented by numerical solution to go beyond the range of validity of the analytical theory. Most of the results of Section VC1 are based on a perturbative analytical approach formulated in the energy domain. Section VC2 utilizes a combination of analytical perturbative and numerical nonperturbative time-domain methods, based on propagation of the system density matrix. Details of our formalism are provided in Refs. 47 and 48 and are not reproduced here. 

In fact, it is true that equation 69-10 represents the least-squares solution to the problem of finding the coefficients of equation 69-3, it is just not obvious from this derivation, based on matrix mathematics. To demonstrate that equation 69-10 is, in fact, a least-squares solution, we have to go back to the initial problem and apply the methods of calculus to the problem. This derivation has been done in great detail [7], and in somewhat lesser detail in a spectroscopic context [8],... 

This analytical method, based on TXRF, enables a large number of trace elements to be determined simultaneously. The range is suitable for different areas of the sea. The motivation to use TXRF resulted mainly from the characteristic features of the method its high detection power, its universal calibration curve, which eliminates the need for matrix-dependent standard samples or standard-addition procedures, the simple preparation of the sample films, and of course the possibility of multielement determination. 

The use of Equation (A. 17) for inversion is conceptually simple, but it is not a very efficient method for calculating the inverse matrix. A method based on use of row operations is discussed in Section A.3. For matrices of size larger than 3 X 3, we recommend that you use software such as MATLAB to find A 1. 

The application of the Chebyshev recursion to complex-symmetric problems is more restricted because Chebyshev polynomials may diverge outside the real axis. Nevertheless, eigenvalues of a complex-symmetric matrix that are close to the real energy axis can be obtained using the FD method based on the damped Chebyshev recursion.155,215 For broad and even overlapping resonances, it has been shown that the use of multiple cross-correlation functions may be beneficial.216... 

The Mahalanobis distance used for multivariate outlier detection relies on the estimation of a covariance matrix (see Section 2.3.2), in this case preferably a robust covariance matrix. However, robust covariance estimators like the MCD estimator need more objects than variables, and thus for many applications with m>n this approach is not possible. For this situation, other multivariate outlier detection techniques can be used like a method based on robustified principal components (Filzmoser et al. 2008). The R code to apply this method on a data set X is as follows ... 

There are essentially two different procedures for robust PCA, a method based on robust estimation of the covariance, and a method based on projection pursuit. For the covariance-based procedure the population covariance matrix X has to be... 

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