Point matching method

In the point matching method (Oguchi, 1973 Bates, 1975) the fields inside and outside a particle are expanded in vector spherical harmonics and the resulting series truncated the tangential field components are required to be continuous at a finite number of points on the particle boundary. Although easy to describe and to understand, the practical usefulness of this method is limited to nearly spherical particles large demands on computer time and uncertain convergence are also drawbacks (Yeh and Mei, 1980). 

The transition matrix relates the expansion coefficients of the incident and scattered fields. The existence of the transition matrix is postulated by the T-Matrix Ansatz and is a consequence of the series expansions of the incident and scattered fields and the linearity of the Maxwell equations. Historically, the transition matrix has been introduced within the null-field method formalism (see [253,256]), and for this reason, the null-field method has often been referred to as the T-matrix method. However, the null-field method is only one among many methods that can be used to compute the transition matrix. The transition matrix can also be derived in the framework of the method of moments [88], the separation of variables method [208], the discrete dipole approximation [151] and the point matching method [181]. Rother et al. [205] foimd a general relation between the surface Green function and the transition matrix for the exterior Maxwell problem, which in principle, allows to compute the transition matrix with the finite-difference technique. 

T.A. Nieminen, H. Rubinsztein-Dunlop, N.R. Heckenberg, Calculation of the T-matrix General considerations and application of the point-matching method, J. Quant. Spectrosc. Radiat. Transfer 79—80, 1019 (2003)... 

Today, there an established software tool set does exist for the primary task, the calculation of modes and the description of field propagation. Approaches based on the finite element method (FEM) and finite differences (FD) are popular since long and can be applied to complex problems . The wave matching method, Green functions approaches, and many more schemes are used. But, as a matter of fact, the more dominant numerical methods are, the more the user has to scrutinize the results from the physical point of view. Recent mathematical methods, which can control accuracy absolutely - at least if the problem is well posed, help the design engineer with this. ... 

As can be deduced, for m > 2, expression (2.67) leads to cross derivatives by x and y, whose evaluation is rather cumbersome. To alleviate this difficulty, only one fictitious point can be considered at each side of the interface and hence only the zero- and first-order jump conditions are implemented. While this notion gives reliable solutions, an alternative quasi-fourth-order strategy has been presented in [28] for the consideration of higher order conditions and crossderivative computation. A fairly interesting feature of the derivative matching method is that it encompasses various schemes with different orders that permit its hybridization with other high-accuracy time-domain approaches. 

We used a bilinear interpolation method to interpolate data collected from various models onto 0.25° X 0.25° grid points. Then, we used an equidistant quantile matching method to correct monthly average data. Finally, we applied the Delta method to generate daily data for the base period and future scenarios. By these means, we obtained daily temperature and precipitation data for 518 0.25° x 0.25° grids of the upper reaches of the Yangtze River in the base period and in the future period under the three scenarios and then input these data into the VIC model. Detailed introductions to the equidistant quantile matching-based... 

This history clearly shows how the development of flash point test methods and testers was matched to the character of the fluid being tested. Today, D56-18T (or its current embodiment) is used to characterize not volatile hydrocarbons, but mixtures of water and alcohol, HCFCs/ HFCs/HFEs (Ref. 3, Chapter 22) and alcohols/hydrocarbons, multiple HCFCs/HFCs/HFEs, chlorinated and brominated hydrocarbons (Ref. 3, Chapters 20 and 21), mixtures of HCFCs/HFCs/HFEs and chlorinated hydrocarbons, and high-boiling alcohols/glycol ethers (Ref. 3, Chapter 14). All of these materials are separately used to clean parts. 

In a titrimetric method of analysis the volume of titrant reacting stoichiometrically with the analyte provides quantitative information about the amount of analyte in a sample. The volume of titrant required to achieve this stoichiometric reaction is called the equivalence point. Experimentally we determine the titration s end point using a visual indicator that changes color near the equivalence point. Alternatively, we can locate the end point by recording a titration curve showing the titration reaction s progress as a function of the titrant s volume. In either case, the end point must closely match the equivalence point if a titration is to be accurate. Knowing the shape of a titration... 

Theoretical studies of the relative stabilities of tautomers 14a and 14b were carried out mostly at the semiempirical level. AMI and PM3 calculations [98JST(T)249] of the relative stabilities carried out for a series of 4(5)-substituted imidazoles 14 (R = H, R = H, CH3, OH, F, NO2, Ph) are mostly in accord with the conclusion based on the Charton s equation. From the comparison of the electronic spectra of 4(5)-phenylimidazole 14 (R2 = Ph, R = R3 = H) and 2,4(5)-diphenylimidazole 14 (R = R = Ph, R = H) in ethanol with those calculated by using ir-electron PPP method for each of the tautomeric forms, it follows that calculations for type 14a tautomers match the experimentally observed spectra better (86ZC378). The AMI calculations [92JCS(P1)2779] of enthalpies of formation of 4(5)-aminoimidazole 14 (R = NH2, R = R = H) and 4(5)-nitroimidazole 14 (R = NO2, R = R = H) point to tautomers 14a and 14b respectively as being energetically preferred in the gas phase. Both predictions are in disagreement with expectations based on Charton s equation and the data related to basicity measurements (Table III). These inconsistencies may be... 

It is desirable for the record to have an objective statement of the nature and degree of color deterioration. The simplest, but least desirable, method is comparison of sample color with color charts or plates such as those used in the Munsell system, Ridgeway s color standards, or the Maerz and Paul dictionary of color. Such a method is limited in value because of the difficulty of obtaining true color matches, and because of variations due to human error. The use of color charts or plates may be much improved in the Munsell system by employing a disk colorimeter (29). Kramer and Smith (21) have pointed out that the results obtained in its application to foods are sometimes difficult to explain and compare, and that the method requires special training of the operator and is tedious and cumbersome. 

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