Problem-solving techniques direct method

The traditional approach for structure solution follows a close analogy to the analysis of single-crystal XRD data, in that the intensities 1(H) of individual reflections are extracted directly from the powder XRD pattern and are then used in the types of structure solution calculation (e.g. direct methods, Patterson methods or the recently developed charge-flipping methodology [32-34]) that are used for single-crystal XRD data. As discussed above, however, peak overlap in the powder XRD pattern can limit the reliability of the extracted intensities, and uncertainties in the intensities can lead to difficulties in subsequent attempts to solve the structure. As noted above, such problems may be particularly severe in cases of large unit cells and low symmetry, as encountered for most molecular solids. In spite of these intrinsic difficulties, however, there have been several reported successes in the application of traditional techniques for structure solution of molecular solids from powder XRD data. 

The last one is based on discretization techniques, received major attention and considered as an efficient solution method. The concept of this approach is to transform the original optimal control problem into a finite dimensional optimization problem, typically a nonlinear programming problem (NLP). Then, the optimal control solution is given by applying a standard NLP solver to directly solve the optimization problem. For this reason, the method is known as a direct method. The transformation of the problem can be made by using discretization technique on either only control variables (partial discretization) or both state and control variables (complete discretization). Based on this con-... 

With the development of high-speed personal computers, it is very convenient to use numerical techniques to solve heat transfer problems. The finite-difference method and the finite-element method are two popular and useful methods. The finite-element method is not as direct, conceptually, as the finite-difference method. It has some advantages over the finite-difference method in solving heat transfer problems, especially for problems with complex geometries. 

Despite the success of direct methods, there are still certain structures that are not readily solved, if at all, by these methods. This may be due to a breakdown in the certainty with which the triplet relationships are developed, or to a breaJidown of the assumption that the atomic position vectors form a set of random variables, so that the probability functions that are derived may no longer be strictly valid. Some new developments for dealing with these problems have focused on improving the techniques for calculating the triplets and including in the direct methods procedure as much structural information as possible, such as... 

The Patterson synthesis (Patterson, 1935), or Patterson map as it is more commonly known, will be discussed in detail in the next chapter. It is important in conjunction with all of the methods above, except perhaps direct methods, but in theory it also offers a means of deducing a molecular structure directly from the intensity data alone. In practice, however, Patterson techniques can be used to solve an entire structure only if the structure contains very few atoms, three or four at most, though sometimes more, up to a dozen or so if the atoms are arranged in a unique motif such as a planar ring structure. Direct deconvolution of the Patterson map to solve even a very small macromolecule is impossible, and it provides no useful approach. Substructures within macromolecular crystals, such as heavy atom constellations (in isomorphous replacement) or constellations of anomalous scattered, however, are amenable to direct Patterson interpretation. These substructures may then be used to solve the phase problem by one of the other techniques described below. 

The methods and techniques of expertise studies have direct value for schema research. Much of what we want to study shows up in intermediate, rather than final, stages of problem solving, and we need procedures that will allow us to gather a wide variety of data. The techniques of Figure 6.1 can be easily adapted for schema study, as will be seen in subsequent chapters. 

A modern computer-controlled diffractometer allows a structural problem to be solved in a matter of days (and in some cases hours). The technique is ideally suited to complicated structures, and has been used quite extensively in the diterpene alkaloid group. The so-called "direct method" (using a single crystal of the free base) has been applied, for example to the C2o-alkaloid veatchine (1) (Pelletier, Mody, and W. H. DeCamp, J. Amer. chem. Soc., 1978, 100,... 

Sometimes in fluid mechanics we may start with these four ideas and the measured physical properties of the materials under consideration and proceed directly to solve mathematically for the desired forces, velocities, and so on. This is generally possible only in the case of very simple flows. The observed behavior of a great many fluid flows is too complex to be solved directly from these four principles, so we must resort to experimental tests. Through the use of techniques called dimensional analysis (Chap. 13) often we can use the results of one experiment to predict the results of a much different experiment. Thus, careful experimental work is very important in fluid mechanics. With the development of supercomputers, we are now able to solve many complex problems mathematically by using the methods outlined in Chaps. 10 and 11, which previously would have required experimental tests. As computers become faster and cheaper, we will probably see additional complex fluid mechanics problems solved on supercomputers. Ultimately, the computer solutions must be tested experimentally. 

On the basis of the results obtained, one can say that the SEFS results are very useful in analyzing the atomic structure of superthin surface layers of matter. But whenever studies of the local atomic structure can be performed by other methods, for example by EELFS, the complexity of the SEFS technique becomes an important disadvantage. However, in the experimental study of atomic PCF s of surface layers of multicomponent atomic systems within the formalism of the inverse problem solution, a complete set of integral equations is necessary to provide mathematical correctness. This set of equations can be solved by the methods of direct solution only. In this case the use of the SEFS method may be a necessary condition for obtaining a reliable result. Besides, the calculations made can be used as a test when studying multicomponent systems. 

In the previous methods, the use of a continuous membrane for measuring cell forces has the disadvantage that the traction forces are convoluted by the observed bead displacements. Because the calculation is not direct, constraints or assumptions are required to solve the inverse problem. A technique to transduce individual traction forces comes from the use of microfabricated cantilevers. The first demonstration of these sensors was a horizontal silicon cantilever that was made with microfabrication techniques (Figure 22.6a). As a cell migrates across the surface, it bends the cantilever under the load of the traction force. Because the sensor is mechanically decoupled from the substrate, the deflection of the cantilever directly reports only the local force. The simple spring equation relates the deflection of the cantilever beam 8 to the cellular traction force ... 

---