Mathematical analysis

Preliminary data analysis carried out for the spectral datasets were functional group mapping, and/or hierarchical cluster analysis (HCA). This latter method, which is well described in the literature,4,9 is an unsupervised approach that does not require any reference datasets. Like most of the multivariate methods, HCA is based on the correlation matrix Cut for all spectra in the dataset. This matrix, defined by Equation (9.1), 

The resulting covariance matrix Cm contains P2 entries, where P is the total number of spectra within the dataset. However, since the matrix is symmetric, only p p - l)/2 spectral distance elements Cm need to be computed. Nevertheless, for large datasets, the size of the covariance matrix may exceed the address space of Pentium IV-class processors, and the computational problems described next are rather unyielding. 

Reasonable noise in the spectral data does not affect the clustering process. In this respect, cluster analysis is much more stable than other methods of multivariate analysis, such as principal component analysis (PCA), in which an increasing amount of noise is accumulated in the less relevant clusters. The mean cluster spectra can be extracted and used for the interpretation of the chemical or biochemical differences between clusters. HCA, per se, is ill-suited for a diagnostic algorithm. We have used the spectra from clusters to train artificial neural networks (ANNs), which may serve as supervised methods for final analysis. This process, which requires hundreds or thousands of spectra from each spectral class, is presently ongoing, and validated and blinded analyses, based on these efforts, will be reported. 

---

Figure 1. shows the measured phase differenee derived using equation (6). A close match between the three sets of data points can be seen. Small jumps in the phase delay at 5tt, 3tt and most noticeably at tt are the result of the mathematical analysis used. As the cell is rotated such that tlie optical axis of the crystal structure runs parallel to the angle of polarisation, the cell acts as a phase-only modulator, and the voltage induced refractive index change no longer provides rotation of polarisation. This is desirable as ultimately the device is to be introduced to an interferometer, and any differing polarisations induced in the beams of such a device results in lower intensity modulation. 

It turns out that many surfaces (and many line patterns such as shown in Fig. XV-7) conform empirically to Eq. VII-20 (or Eq. VII-21) over a significant range of r (or a). Fractal surfaces thus constitute an extreme departure from ideal plane surfaces yet are amenable to mathematical analysis. There is a considerable literature on the subject, but Refs. 104-109 are representative. The fractal approach to adsorption phenomena is discussed in Section XVI-13. 

A detailed mathematical analysis has been possible for a second situation, of a wetting meniscus against a flat plate, illustrated in Fig. X-16b. The relevant equation is [226]... 

This algebra implies that in case of Eq. (111) the only two functions (out of n) that flip sign are and because all in-between functions get their sign flipped twice. In the same way, Eq. (112) implies that all four electronic functions mentioned in the expression, namely, the jth and the (j + 1 )th, the th and the (/c -h 1 )th, all flip sign. In what follows, we give a more detailed explanation based on the mathematical analysis of the Section Vin. 

A formal mathematical analysis of the flow in the concentric cylinder viscometer yields the following relationship between the experimental variables and the viscosity ... 

We begin the mathematical analysis of the model, by considering the forces acting on one of the beads. If the sample is subject to stress in only one direction, it is sufficient to set up a one-dimensional problem and examine the components of force, velocity, and displacement in the direction of the stress. We assume this to be the z direction. The subchains and their associated beads and springs are indexed from 1 to N we focus attention on the ith. The absolute coordinates of the beads do not concern us, only their displacements. 

Our approach to the problem of gelation proceeds through two stages First we consider the probability that AA and BB polymerize until all chain segments are capped by an Aj- monomer then we consider the probability that these are connected together to form a network. The actual molecular processes occur at random and not in this sequence, but mathematical analysis is feasible if we consider the process in stages. As long as the same sort of structure results from both the random and the subdivided processes, the analysis is valid. 

Hardness is a measure of a material s resistance to deformation. In this article hardness is taken to be the measure of a material s resistance to indentation by a tool or indenter harder than itself This seems a relatively simple concept until mathematical analysis is attempted the elastic, plastic, and elastic recovery properties of a material are involved, making the relationship quite complex. Further complications are introduced by variations in elastic modulus and frictional coefficients. 

Table 10-56 gives values for the modulus of elasticity for nonmetals however, no specific stress-limiting criteria or methods of stress analysis are presented. Stress-strain behavior of most nonmetals differs considerably from that of metals and is less well-defined for mathematic analysis. The piping system should be designed and laid out so that flexural stresses resulting from displacement due to expansion, contraction, and other movement are minimized. This concept requires special attention to supports, terminals, and other restraints. 

A mathematical analysis of the action in Kady and other colloid mills checks well with experimental performance [Turner and McCarthy, Am. Inst. Chem. Eng. J., 12(4), 784 (1966)], Various models of the Kady mill have been described, and capacities and costs given by Zimmerman and Lavine [Co.st Eng., 12(1), 4-8 (1967)]. Energy requirements differ so much with the materials involved that other devices are often used to obtain the same end. These include high-speed stirrers, turbine mixers, bead mills, and vibratoiy mills. In some cases, sonic devices are effec tive. 

The well-known difficulty with batch reactors is the uncertainty of the initial reaction conditions. The problem is to bring together reactants, catalyst and operating conditions of temperature and pressure so that at zero time everything is as desired. The initial reaction rate is usually the fastest and most error-laden. To overcome this, the traditional method was to calculate the rate for decreasingly smaller conversions and extrapolate it back to zero conversion. The significance of estimating initial rate was that without any products present, rate could be expressed as the function of reactants and temperature only. This then simplified the mathematical analysis of the rate fianction. 

In gradientless reactors the catalytic rate is measured under highly, even if not completely uniform conditions of temperature and concentration. The reason is that, if achieved, the subsequent mathematical analysis and kinetic interpretation will be simpler to perform and the results can be used more reliably. The many ways of approximating gradientless operating conditions in laboratory reactors will be discussed next. 

The feed-back design (Figure 6.3.3 on the next page) was a 2-level, 6-variables central composite plan that required 2 = 64 experiments for the full replica. A 1/4 replica consisting of 16 experiments was made with an additional centerpoint. This was repeated after every 3 to 4 experiments to check for the unchanged condition of the catalyst. The execution of the complete study required six weeks of around the clock work. In the next six weeks, mathematical analysis and model-building was done and some additional check experiments were made. 

Many of the better known shortcut equipment design methods have been derived by informed assumptions and mathematical analysis. Testing in the laboratory or field was classically used to validate these methods but computers now help by providing easy access to rigorous design calculations. 

Crystallography is a very broad science, stretching from crystal-structure determination to crystal physics (especially the systematic study and mathematical analysis of anisotropy), crystal chemistry and the geometrical study of phase transitions in the solid state, and stretching to the prediction of crystal structures from first principles this last is very active nowadays and is entirely dependent on recent advances in the electron theory of solids. There is also a flourishing field of applied crystallography, encompassing such skills as the determination of preferred orientations, alias textures, in polycrystalline assemblies. It would be fair to say that... 

Modeling the Release of Chemicals (predicting the path, the effect, and the area of impact of the chemical release using mathematical analysis)... 

For the force we have a model, Eq. (14.101), but we also need a formula for the force The following model is based on our own idea and is not presented elsewhere. In Section 14.3.2 we will briefly discuss another way of modeling the force but without any detailed mathematical analysis. 

To constitute the We number, characteristic values such as the drop diameter, d, and particularly the interfacial tension, w, must be experimentally determined. However, the We number can also be obtained by deduction from mathematical analysis of droplet deforma-tional properties assuming a realistic model of the system. For a shear flow that is still dominant in the case of injection molding, Cox [25] derived an expression that for Newtonian fluids at not too high deformation has been proven to be valid ... 

No exact mathematical analysis of potential attenuation for all structures has yet been developed. Some indicative analysis has been achieved for a buried pipelinewhich is perhaps the simplest case. 

The task of finding and classifying periodic solutions is in principle a straightforward one if a formal mathematical analysis proves too difficult, one may always... 

This finite difference obviously becomes increasingly less important in the limit of very long strings. For purposes of mathematical analysis, Ku(s) can be treated as being effectively independent of U. 

Consider the apparatus shown in Figure 14-6. The equipment is similar to that shown in Figure 14-4 except a fluorescent screen within the tube reveals the trajectory of the particles that pass through the slot in the positive electrode. When a magnetic field is added, the electron trajectory is curved. A mathematical analysis of the curvature permits an interpretation of this experiment that leads to a determination of e/m. 

W. Kudin, Principles of Mathematical Analysis, McGraw-Hill Book Co., New York, 1953. 

McAlevy (M3, M4) developed an approximate mathematical analysis for the ignition of a pyrolyzing fuel in an oxidizing environment as shown in Fig. 5. Hermance (H4, H5) expanded the analysis by incorporating the contributions which were previously omitted. However, the effects of natural... 

---