Formal analysis

A somewhat similar problem arises in describing the viscosity of a suspension of spherical particles. This problem was analyzed by Einstein in 1906, with some corrections appearing in 1911. As we did with Stokes law, we shall only present qualitative arguments which give plausibility to the final form. The fact that it took Einstein 5 years to work out the bugs in this theory is an indication of the complexity of the formal analysis. Derivations of both the Stokes and Einstein equations which do not require vector calculus have been presented by Lauffer [Ref. 3]. The latter derivations are at about the same level of difficulty as most of the mathematics in this book. We shall only hint at the direction of Lauffer s derivation, however, since our interest in rigid spheres is marginal, at best. 

Requirements for Analysis No formal analysis of adequate flexibility is required in systems which (1) are dupKcates of successfully operating instahations or replacements without significant change of systems with a satisfactory service record (2) can readily be judged... 

Throughout this section the hydronium ion and hydroxide ion concentrations appear in rate equations. For convenience these are written [H ] and [OH ]. Usually, of course, these quantities have been estimated from a measured pH, so they are conventional activities rather than concentrations. However, our present concern is with the formal analysis of rate equations, and we can conveniently assume that activity coefficients are unity or are at least constant. The basic experimental information is k, the pseudo-first-order rate constant, as a function of pH. Within a senes of such measurements the ionic strength should be held constant. If the pH is maintained constant with a buffer, k should be measured at more than one buffer concentration (but at constant pH) to see if the buffer affects the rate. If such a dependence is observed, the rate constant should be measured at several buffer concentrations and extrapolated to zero buffer to give the correct k for that pH. 

A formal analysis of the behavior of general reversible systems will be given in section 4.3. An explicit example of a two-dimensional reversible universal computer... 

Before doing any formal analysis, we state a few qualitative features of each type of controller. This is one advantage of classical control. We can make fairly easy physical interpretation of the control algorithm. The analyses that come later will confirm these qualitative observations. 

We expect a system with only a proportional controller to have a steady state error (or an offset). A formal analysis will be introduced in the next section. This is one simplistic way to see why. Let s say we change the system to a new set point. The proportional controller output, p = ps + Kce, is required to shift away from the previous bias ps and move the system to a new steady state. For p to be different from ps, the error must have a finite non-zero value.3... 

The sign of the rate of change in the error could be opposite that of the proportional or integral terms. Thus adding derivative action to PI control may counteract the overcompensation of the integrating action. PD control may improve system response while reducing oscillations and overshoot. (Formal analysis later will show that the problem is more complex than this simple statement.)... 

It was noted early by Reed and others that the IETS spectrum could exhibit both absorption and emission peaks - that is, the plots of Fig. 9 could have positive excursions and negative excursions called peaks and dips. The simple analysis suggested in Fig. 9 implies that it should always be absorptive behavior, and therefore that there should always be a peak (a maximum, an enhancement) in the IETS spectrum at the vibrational resonances. It has been observed, however, that dips sometimes occur in these spectra. These have been particularly visible in small molecules in junctions, such as in the work of van Ruitenbeek [92, 109] (Fig. 12). Here, formal analysis indicates that, as the injection gap gets smaller, the existence of an inelastic vibrational channel does not contribute a second independent channel to the transport, but rather opens up an interference [100]. This interference can actually impede transport, resulting in a dip in the spectrum. Qualitatively, this occurs because the system is close to an electronic resonance without the vibrational coupling the conductance is close to g0, and the interference subtracts from the current. 

The data from toxicology studies should always be examined before any formal analysis Such examinations should be directed to determining if the data are suitable for analysis, and if so what form the analysis should take (see Figure 22.2.) If the data as collected are not suitable for analysis, or if they are only suitable for low-... 

Based on the accessibility of high-quality experimental information, we now focus on aspects of model interrogation and analysis. The question how cells actually control and distribute their flux under different conditions requires a mathematical and formal approach to metabolic regulation. The knowledge obtained by quantitative experiments must be, in the sense of Section II.B, encoded into a mathematical system, scrutinized utilizing the tools of formal analysis, and eventually decoded back into predictions about the natural system. 

The established tools of nonlinear dynamics provide an elaborate and versatile mathematical framework to examine the dynamic properties of metabolic systems. In this context, the metabolic balance equation (Eq. 5) constitutes a deterministic nonlinear dynamic system, amenable to systematic formal analysis. We are interested in the asymptotic, the linear stability of metabolic states, and transitions between different dynamic regimes (bifurcations). For a more detailed account, see also the monographs of Strogatz [290], Kaplan and Glass [18], as well as several related works on the topic [291 293],... 

Aiming at a more formal analysis, the asymptotic stability of a steady-state value S° of a metabolic system upon an infinitesimal perturbation is determined by linear stability analysis. Given a metabolic system at a positive steady-state value... 

Providing a more formal analysis, several objective measures to rank the parameters according to their impact on the stability can be utilized. Possible measures of dependency are ... 

Requirements for Analysis No formal analysis of adequate flexibility is required in systems which (1) are duplicates of successfully operating installations or replacements without significant change of systems with a satisfactory service record (2) can readily be judged adequate by comparison with previously analyzed systems or (3) are of uniform size, have no more than two points of fixation, have no intermediate restraints, and fall within the limitations of empirical Eq. (10-100) ... 

Complete formal analysis of the recoil corrections in the framework of the relativistic two-particle equations, with derivation of all relevant kernels, perturbation theory contributions, and necessary subtraction terms may be performed along the same lines as was done for hyperfine splitting in [3]. However, these results may also be understood without a cumbersome formalism by starting with the simple scattering approximation. We will discuss recoil corrections below using this less rigorous but more physically transparent approach. 

Kliiver s emphasis on form constants in his human mescaline work and his formal analysis of emotional behavior in his monkey lesion studies are scientifically exemplary. Although individuals vary widely in exactly... 

This situation is found for the binding of ligands to chymotrypsin, which exists in two conformational states. Only one of these states binds aromatic substrates. It may be shown from the formal analysis to be given in section 6 that... 

The ideal mutation is a nondisruptive deletion that is, one that simply removes an interaction without causing a disruption or reorganization of structure. A formal analysis shows that measurements on nondisruptive deletants provide information on the energetic changes in the wild-type enzyme.11... 

Both Craig and Weiss (1971) and Bieri (1971) stress that analysis of the He anomalies must be based on comparison with those of Ne and Ar. Craig and Weiss (1971) made a formal analysis expressing each of AHe, ANe, and AAr as a linear sum of terms in AP, AT, and Aa (Section 4.1). A convenient graphical presentation (Figure 4.3) is the relation between AHe-ANe and ANe-AAr. Since pressure variations... 

Both formal analysis and computational developments associated with DFT can be carried over intact to nDFT. For example, the exact two-particle ground-state density, no(x), can be determined through a constrained search [34] for that many-particle, properly symmetrized or antisymmetrized wave function, with symmetry imposed with respect to ordinary particles, which yields n0 and also minimizes the many-particle energy, T + Vpp, where Vpp denotes the interparticle interaction in two-particle space. Essentially any method developed within a single-particle application of DFT for the study of electronic structure can, with appropriate technical modifications, be extended to two-, or rc-particle states. The use of multiple-scattering theory to calculate fully correlated two-particle densities in solids will be given in a future publication. 

---