Selecting a solution

TREOR and ITO programs have an option to input intensity for each observed diffraction peak but they are included only for informative purposes and are never analyzed by the code. Nevertheless, it is often helpful to see the intensity next to indices when analyzing the results, especially for those Bragg peaks, which do not fit into the selected lattice. 

All three programs TREOR, DICVOL and ITO allow optional input of the information about the measured gravimetric density and formula weight in order to estimate the number of formula units expected in the found unit cell (see section 6.3 in Chapter 6). The latter should be an integer number compatible with the unit cell symmetry, e.g. in a primitive monoclinic lattice it normally should be a multiple of 2 or 4. The agreement between the number of formula units in the unit cell and lattice symmetry may be used as an additional stipulation when selecting the most probable solution. 

Automatic indexing programs usually produce more than one solution and the following criteria should be considered when deciding which one is the best, or in other words, which solution we believe represents the true unit cell  

The correct solution must result in a high figure of merit (although not necessarily the highest) and most, if not all diffraction peaks should be indexed. Unindexed Bragg reflections must be explained, for example, by low intensity, by noticeably different peak widths, or they should be identified as corresponding to an impurity phase. 

As is stated in the DICVOL manual ... a solution from the first 20 (or N) lines. .. must index the complete powder diffraction pattern. .. . The best way to evaluate the quality of the indexing is to perform lattice parameters refinement and to calculate a figure of merit using all measured and indexed Bragg peaks. 

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Although froth flotation is recognized as the best available fine coal cleaning technique, it becomes iueffective when the particle size is much smaller than 0.1 mm or when the feed contains a large amount of clay, resulting iu low coal recovery or poor selectivity. A solution to these problems is the use of modified flotation devices. 

The circulation rates for amine systems can be determined from the acid gas flow rates by selecting a solution concentration and an acid gas loading. 

The first term measures the difference between the data and the fit, KF. The second term is a Tikhonov regularization and its amplitude is controlled by the parameter a. The effect of this regularization term is to select a solution with a small 2-norm 11 F 2 and as a result a solution that is smooth and without sharp spikes. However, it may cause a bias to the result. When a is chosen such that the two terms are comparable, the bias is minimized and the result is stable in the presence of noise. When a is much smaller, the resulting spectrum F can become unstable. 

The other solutions to Eq. (3.13) correspond to stationary points where the function is increased in some directions and reduced in others. For example, if we select a solution in the region X2 < p < X3 then the step is toward the gradient of the first two modes and opposite the gradient of all higher modes. The second-order change in the function may be written... 

Based on Tikhonov s theorem, we know that the operator A is a continuous one on the correctness set C. Therefore, we conclude from (2.22) that the quasi-solution is a continuous function of d. Note that this property holds only in the correctness set C. If one selects a solution, m, from outside the correctness set, it may be no longer a continuous function of the data (see Figure 2-1). 

We have introduced a stabilizing functional under the condition that it selects a compact subset Me from a metric space of the model parameters. Thus, we can say that a stabilizer from a set of possible solutions, Qs, selects a solution, which at the same time belongs to the correctness set Me- Figure 2-3 helps to explain this role of the stabilizing functional. 

A detailed look at the evolution of soil-moisture chemistry was reported by Sears (1976). In his study Sears assumed the average composition of precipitation shown in Table 8.7. Table 8.7 also lists analyses of the soil moisture he collected from suction lysimeters at 1- and 3-m depths in respective B- and C-horizon soils formed by the weathering of underlying sandy dolomite. The 1-m sample is chiefly a Na -NOj water, with the nitrate probably from fertilizer. The TDS is about 70 mg/L at 1 m and has increased to 500 mg/L at the 3-m depth. In order to explain changes occurring between the 1- and 3-m depth, it is useful to select a solute we can assume to be practically un-reactive in the soil. The best common species for this purpose is probably Cl, with which we can then compare other species concentrations. Relative increases from 1- to 3-m depth are shown in the third column. Increases compared to chloride are given in the fourth column. 

Once the DM has classified the objective functions, (s)he can decide how many Pareto optimal solutions (between one and four) based on this information (s)he wants to see and compare. Then, as many scalarized problems are formed and solved and the new solutions are shown to the DM together with the current solution. If the DM has found the most preferred solution, the solution process stops. Otherwise, the DM can select a solution as a starting point of a new classification or ask for a desired number of intermediate (Pareto optimal) solutions between any two solutions generated so far. The DM can also save any interesting solutions to a database and return to them later. All the solutions considered are Pareto optimal. For details of the algorithm, see Miettinen and Makela (2006). 

Quality-Improvement Process. If the threshold is not met, a trend is identified, or a pattern is observed, a quality-improvement opportunity exists. A team could be formed to review the indicator, examine the process that the indicator measured, define the problem encountered, identify ways to solve the problem, and select a solution. An action plan will then be developed to implement this solution. 

One way to work with new populations of respondents is to listen to them, understand their feelings, and then select a solution which works, ideally the optimum solution. The two halves are the approach we call Empathy (getting to the respondent s heart) and Experiment (understanding what works). 

The elimination of c from Eq. 37 means that the iteration process does not select a solution with a given velocity c but a solution with a given integrated amplitude... 

Select a solution among the best solutions randomly... 

Code Implementation. Attending to the error vectors in Table 1 and Table 2, and selecting a solution with smallest Hamming weight in H, one possible solution is ... 

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