Odd-number sampling

For the remaining subsections of Section 5.3, the water-methanol data will be considered split such that the 6 odd-numbered samples of the 11 denote the calibration set and the remaining 5 even-numbered samples compose the validation set. In some situations as noted, the calibration set consists of all 11 samples. 

FIGURE 5.7 Graphical displays for the methanol model at 1452,1932, 2072, and 2274 nm with a nonzero intercept using all 11 calibration samples. The RMSEC is 0.16% methanol, (a) Calibration residual plot, (b) A plot of estimated values against the actual values for the calibration samples the drawn line is the line of equality, (c) Validation residual plot after the 11 samples were split to 6 calibration (odd-numbered samples) and 5 validation (even-numbered samples). 

Various methods of integration, (a) Trapezoidal-1, (b) Tropezoidal-2, (c) Simpson s method, and (d) Odd-number sampling. 

The following method of odd-number sampling was developed by Wedepohl [6,40,41] ... 

The results of a unit step function calculated by these integration methods are shown in Figure 2.61 [41]. In the calculations, the weighting function is included. Table 2.8 shows a comparison of accuracy between the various methods. From the results, it is obvious that the odd-number sampling method (d) is the most accurate and efficient. 

Application of the odd-number sampling modified Laplace transform (MET)... 

Within the VGO saturates, distribution of paraffins, isoparaffins, and naphthenes is highly dependent on the petroleum source. The naphthenes account for roughly 60% of the saturates in a normal cmde oil. However, samples can be found having paraffins from <20 to >80%. In most samples, the / -paraffins from C2Q—are still present in sufficient quantity to be detected as distinct peaks in gc analyses. Some cmde oils show a nearly symmetric pattern of peaks such that each carbon number is present in regular progression up to a maximum around C -j. Other cmde oils show a similar distribution, but have preference for odd-numbered alkanes. Both the distribution and the selectivity toward odd-numbered hydrocarbons are considered to reflect differences in petrogenesis of the cmde oils. Although / -paraffins are distinct in the gc, these usually account for only a few percent of the saturates measured by gc. 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

Median. When all the numbers in a group are arranged in a ranked order (that is, from smallest to largest), the median is the middle value. If there is an odd number of values in a group, then the middle value is obvious (in the case of 13 values, for example, the seventh largest is the median). When the number of values in the sample is even, the median is calculated as the midpoint between the (N/2)th and the ([/Y/2] + l)th number. For example, in the series of numbers 7, 12, 13, 19 the median value would be the midpoint between 12 and 13, which is 12.5. 

The median of an even number of results is nothing but the average of the two middle values provided the results are listed in order whereas for an odd number of results the median is the middle value itself. However, the mean and the median are exactly identical in the case of a truly symmetrical distribution. In short, median is an useful measure specifically when dealing with very small samples. 

In this equation, the image value im is the mth sample of the smeared version of ok. We transform the subscripts in an obvious way that recognizes the commutative property of convolution and considers only spread functions that are nonvanishing over some finite domain having an odd number of points. We may then write... 

While warming the sample, the intensity of the odd number hyperfine structures increased relative to the octet, but no definite assignments of these odd line spectra to specific radical structures were attempted. 

Normally, odd-numbered fatty acids are used as internal standards. While the use of internal standards ensures the correctness of the extraction procedure, it does not guarantee the completeness of extraction for different fatty acids. Due to this reason, a comparison between the methods is essential to truly determine the efficacy of extraction. Chavarri et al. (1997) compared two sample preparation procedures. The first method was the direct method developed by de Jong and Badings (1990), described above. The second method involved saponification with TMAH as described by Martin-Hemandez et al. (1988) and the formation of methyl esters in the injector prior to analysis. The authors found that separation of the FFAs from the triglycerides prior to derivatization improved the analysis. Another comparative study by Ardo and Polychroniadou (1999) reported that the saponification method described above (Martin-Hernandez et al., 1988) was found suitable for both low and high FFA levels in cheese. 

The //-alkanes usually range in chain length from 21 to 31 or 33 carbons. Hydrocarbons with fewer than 20 carbons commonly occur as pheromones, defensive compounds and intermediates to pheromones and defensive compounds, but their volatility makes them unsuited to function as cuticular components, n-Alkanes have been found on almost every insect species analyzed, and can range from less than one percent of the total hydrocarbons, as in tsetse flies (Nelson and Carlson, 1986 Nelson et al., 1988) to almost all of the hydrocarbon fraction, as in the adult tenebrionid beetle, Eurychora sp. (Lockey, 1985). Depending upon the species, they can consist of essentially only one major component, such as n-pentacosane in the American cockroach, Periplaneta americana (Jackson, 1972) to a series of //-alkanes, such as the series from C23 to C33 in the housefly, Musca domes-tica (Nelson et al., 1981), with trace amounts to C37 (Mpuru et al., 2001). In all cases, the odd-numbered alkanes predominate, due to their formation from mostly two carbon units followed by a decarboxylation (Blomquist, Chapter 3, this book). Small amounts of even-numbered carbon chain //-alkanes often occur, and presumably arise from chain initiation with a propionyl-CoA rather than an acetyl-CoA. Occasionally, gas chromatographic analyses reveal similar amounts of even-numbered chain //-alkanes and odd-numbered chain components. This is a red flag that the samples must be checked for contamination. 

Mass spectrometry is used to measure the molecular mass of a compound and provides a method to obtain the molecular formula. It differs from the other instrumental techniques presented thus far because it does not involve the interaction of electromagnetic radiation with the compound. Instead, molecules of the compound being studied are bombarded with a high-energy beam of electrons in the vapor phase. When an electron from the beam impacts on a molecule of the sample, it knocks an electron out of the molecule. The product, called the molecular ion (represented as A/f), has the same mass as the original molecule but has one less electron. It has both an odd number of... 

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