Mathematical arguments

Equation (2.6) is called the Fokker-Planck equation (FPE) or forward Kolmogorov equation, because it contains time derivative of final moment of time t > to. This equation is also known as Smoluchowski equation. The second equation (2.7) is called the backward Kolmogorov equation, because it contains the time derivative of the initial moment of time to < t. These names are associated with the fact that the first equation used Fokker (1914) [44] and Planck (1917) [45] for the description of Brownian motion, but Kolmogorov [46] was the first to give rigorous mathematical argumentation for Eq. (2.6) and he was first to derive Eq. (2.7). The derivation of the FPE may be found, for example, in textbooks [2,15,17,18],... 

Having briefly noted the historical highlights of the WSL effect, we now examine the basic mathematical argument for its existence, namely, the idea that the energy spectrum of an infinite crystal is discretized by an applied held. For the present discussions, we assume that the applied field is linear, with its strength given by its gradient 7. 

There is a considerable literature [10-13] devoted to finding approximate formulas for the frequency of the simple pendulum for non-zero amplitudes, usually based on mathematical arguments designed to approximate elliptic functions. 

For values of, (Xm(n)), where n[Pg.165]

The treatment which follows is of a largely qualitative nature as it seems likely that readers will fall into two classes. On the one hand there will be those who have already, at some point in their careers, familiarised themselves with the basic mathematical arguments which lead to the results quoted here. On the other hand are those who do not wish to be encumbered with lengthy algebraic manipulations but who wish to understand the basic physical mechanisms which are responsible for these results. 

The three characteristics, or conditions, as they are called, of a particular diffusion process cannot be rediscovered by mathematical argument applied to the differential equation. To get at the three conditions, one has to resort to a physical understanding of the diffusion process. Only then can one proceed with the solution of the (now) total differential equation (4.42) and get the precise functional relationship between concentration, distance, and time. 

For simple shear experiments, somewhat lengthy mathematical arguments indicate that the stress tensor for second-order fluids is given by (2)... 

In the present chapter we will attempt to systematize and analyze the fundamentals of the various approaches. In doing so, we will try to attain a level that is high enough to cover the essential physical phenomena, but will avoid advanced mathematical arguments. Roughly speaking this will be done in the above sequence. Readers who are not interested in formalistic approaches can therefore start later in this chapter, and conversely. 

Open-tubular columns behave in exactly the same way as packed columns with respect to pressure. The same mathematical arguments can be educed which results in the modified form of the equation shown in Eq. (2). As the column is geometrically simple, the respective functions of k can also be explicitly developed. 

Then a chapter introducing the concepts for the reader who has not encountered subdivision curves and surfaces before and how the rules are described in ways that we can apply mathematical arguments to. 

Primarily, secrecy schemes were considered, and no formal notion of security existed. Some even thought it impossible that such a notion could exist, the more so because several schemes had been broken for whose security mathematical arguments had been given. However, these arguments had only referred to some aspects of security, e.g., how many keys were possible (see, e.g., [DiHe76]). 

Figure 12. Two ways of experimenting with eam stem. . One involves the conventional process of formulating hypotheses, de.signing and conducting e.speriments, and analysis and interpretation of results. The second involves the abstraction of the system into a model, application of mathematical argument, and interpretation of mathematical conclusions (Van Dyne, 1995).

It is immediately apparent that a theory like transition-state theory is making no pretensions at stating and describing the underlying principles of the behavior of the system. In any serious analysis in terms of the deeper and more fundamental laws of physics (of quantum mechanics, in particular) the further assumptions in its derivation are arbitrary, artificial, and somewhere between wildly simplistic and quite unsound. Nevertheless, the theory is typically introduced via a complex mathematical argument in which it is derived using a series of assumptions and approximations from the supposedly underlying equations of quantum theory and/or statistical mechanics. 

It may well seem that the introduction of the concept of hybridization to account for the valency of the carbon atom is both arbitrary and artificial, and this would indeed be the case if it had no other basis. As with so many other aspects of the application of quantum mechanics to valency theory, however, the justification is mathematical, and here, where we are concerned only with those aspects of valency theory which have a direct bearing on crystal structure, we shall unfortunately often be compelled to quote conclusions without seeking a rigorous justification for the mathematical arguments the reader is referred to the many works on the theory of valency already available. 

Here the sum over segments is not time-dependent since the only relative segmental motion allowed is rotation. The sum is invariant to any rotation of the sphere. A formal mathematical argument produces the same result (Pecora, 1968), but it is clear on physical grounds that the model sphere looks the same to the light wave in any orientation. Thus, Eq. (8.6.6) becomes... 

We next discuss why some steady states are stable and others are unstable. This discussion comes in two parts. First we present a plausibility argument and develop some physical intuition by constructing and examining van Heerden diagrams (251. In the next section we present a rigorous mathematical argument, which has wide applicability in analyzing the. stability of any system, described by differential, equations.. 

Some technical readers may like to see a proof that the Ford circles representing any two different fractipns cannot overlap. (In the extreme case, they may be tangent.) The mathematical argument follows that of Rademacher. 

Otto [26] and later Kudin and Scusseria [27] realized that the major problem for directly including the field in an electronic-structure calculation is related to the fact that the field destroys the periodicity. On the other hand, as mentioned above, both mathematical arguments and actual calculations have found that the charge distribution inside an extended system remains periodie also in the presence of an external field. Therefore, Otto sought a separation of the form... 

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