Direct method, defined

An alternative to obtaining 0 directly involves defining some more convenient shape-dependent function, and an early but still very practical method is the following. We define a shape-dependent quantity as S = dsldg, as indicated in Fig. 11-16, de is the equatorial diameter and ds is the diameter measured at a distance de up from the bottom of the drop. The hard-to-measure size parameter h in Eq. 11-17 is combined with 0 by defining the quantity H = -0(defb). Thus... 

A direct method for introduction of a C-21 acetoxyl group into a 20-keto-pregnane is by reaction with lead tetraacetate at room temperature. Although originally the reaction carried out in hot acetic acid gave low yields, a careful study by Henbest has defined conditions so that yields as high as 86 % can be obtained at room temperature. The preferred solvent is 5 % methanol in benzene, with boron trifluoride etherate as catalyst. With either methanol or benzene, the yield is less than 4%. 

It may happen that AH is not available for the buffer substance used in the kinetic studies moreover the thermodynamic quantity A//° is not precisely the correct quantity to use in Eq. (6-37) because it does not apply to the experimental solvent composition. Then the experimentalist can determine AH. The most direct method is to measure AH calorimetrically however, few laboratories Eire equipped for this measurement. An alternative approach is to measure K, under the kinetic conditions of temperature and solvent this can be done potentiometrically or by potentiometry combined with spectrophotometry. Then, from the slope of the plot of log K a against l/T, AH is calculated. Although this value is not thermodynamically defined (since it is based on the assumption that AH is temperature independent), it will be valid for the present purpose over the temperature range studied. 

The first order (i.c. ]> 1) approximation of the CML system defined by equation 8.44 (using either of the two methods defined above) is given by an elementary fc = 2, r = 1 CA. Since there are only 32 such rules, the particular CA rule corresponding to a CML system with parameters e and s may be found directly by calculating the outcome of each of the five possible local states. Looking at the first-order step function fi x) in equation 8.47, we can identify the absorbing state X = X with the CA state ct = 0, and x = 1/2 with a = 1. 

Since Sg satisfies a system of equations identical with that satisfied by X, except that Rg replaces H, a standard method for improving the result of a direct method is to do the replacement and solve as before for the correction. This obtained correction will not, of course, be the true Sg) but only an approximation Sg, and it will have been obtained as a result of a set of operations that is equivalent to the formation of CaRg, in accordance with Eq. (2-3). The Ca = C is not known explicitly, but is defined implicitly (see the methods of factorization below). 

It is important to distinguish between the concept of method and the concept of technique . Method defines the aggregative state of the original material (e.g., liquid or gaseous state) from which crystals form directly. Each method is employed by means of several techniques. ... 

Figure 13. Principle of direct methods using triplet relations. As shown in the lower right-hand image the trial structure eonsists of atoms which are located at the eomers of the unit eell. Aeeording to the Z2 formula (Sayre equation) a strict phase relation exists within a eertain set of three reflections (a triplet) with large normalized structure factor amplitudes Eu. Sueh a triplet or origin invariant sum is defined as hiEli + + h k h = 0 or hiEli +...

Note that traditional direct methods are wrong 2 out of 3 times whereas likelihood is always correct. The indications for reflection (1) are very strong so we define the phase of (1) as 180°. This becomes node 1 of the phasing tree and we now permute the reflections listed in Table 6. This gives 4 x2=32 nodes i.e. nodes 2-33. The resulting EEGs after entropy maximisation are given in Table 7. 

Hence, two methods are available that can be applied to follow nanoparticles formation and growth (i) an indirect method that utilizes the consumption of molecular hydrogen pressure versus time and (ii) a direct method that follows the loss of precursor by the 1 1 conversion of its cyclo-octadiene ligand to cyclo-octane by GLC measurements. The mechanism developed by Watzky and Finke suggests that the nanoparticles act as Hving-metal polymers -a concept that could be used to obtain particles with defined sizes simply by adding the appropriate amounts of catalyst precursors [32]. 

A simple and direct method of contact angle measurement has also been proposed (Yamaki and Katayama, 1975 Carroll, 1976) by observing the shape of the liquid droplet attached to a single fiber, the so-called droplet aspect ratio method . The liquid is assumed to form a symmetrical droplet about the fiber axis as shown schematically in Fig. 2.21. Neglecting the effect of gravity, the droplet shape can be defined by the following expression ... 

The phase problem of X-ray crystallography may be defined as the problem of determining the phases ( ) of the normalized structure factors E when only the magnitudes E are given. Since there are many more reflections in a diffraction pattern than there are independent atoms in the corresponding crystal, the phase problem is overdetermined, and the existence of relationships among the measured magnitudes is implied. Direct methods (Hauptman and Karle, 1953) are ab initio probabilistic methods that seek to exploit these relationships, and the techniques of probability theory have identified the linear combinations of three phases whose Miller indices sum to... 

In engineering applications it is often convenient to obtain integral representations which directly involve the field and its fluxes, rather than equations for single- or double-layer densities. This methodology is commonly called the direct method. For Poisson s equation this can be done using the Green s identities for scalar fields. As we already know, Poisson s equation is widely used in transport phenomena and polymer processing, and it is defined as,... 

Whereas the radioactivity measurement alone does not allow distinguishing between drug and metabo-lite(s), samples obtained from the described studies should be also used for standard determination of the drug and its known metabolites, receiving information about the drag and the known metabolite kinetics directly. The gap between radioactivity concentrations and the concentrations determined by direct bioana-lytical methods defines the contribution of unknown metabolites. 

In the classic Newton method, the Newton direction is used to update each previous iterate by the formula xfe+1 = x + pfe, until convergence. The reader may recognize the one-dimensional version of Newton s method for solving a nonlinear equation f(x) = 0 x +1 = xk — f(xk)/f (xk). The analogous iteration process for minimizing f x) is x +1 — xk — f xk)lf"(xk). Note that the one-dimensional search vector, -f xit)lf"(.xk), is replaced by the Newton direction -Hk lgt in the multivariate case. This direction is defined for nonsingular Hk. When x0 is sufficiently close to a solution x, quadratic convergence can be proven for Newton s method.3-6 That is, a constant 3 exists such that... 

The best and most direct methods to observe the hydration of a macromolecule are X-ray and neutron diffraction analyses carried out at high resolution to better than 1.8 A. In the crystals of proteins, the macromolecules are heavily hydrated so that between 20Vo and 909o of the total volume are solvent. In fact, despite their well-defined crystal morphology, crystalline proteins more resemble concentrated protein solutions than the solid state. 

---