Gauge functions

The left-hand side of (28) is just the gauge function in (23), and we can therefore say that the vector potential specified by,... 

The function G e) is called the gauge function . In the sense of a test function, the gauge function is in principle arbitrary, but not all choices of G yield a finite measure //. In order to find a suitable gauge function the following expansion turned out to be useful ... 

The gauge function (8.2.5) contains a logarithmic term, di = —1.2, which guarantees that, for the reduced fractal, do = 1 and a zero measure are compatible with each other. 

The semi-classical equations of motion obtained above involve only the transverse adiabatic vector potential which is, by definition, independent of the choice of gauge functions/(q) and g(q). The (Aj -f A2)/2M term in the potential is also independent of those two arbitrary functions. The locally quadratic approach to Gaussian dynamics therefore gives physically equivalent results for any choice of /(q) and g(q). The finding that the locally quadratic Hamiltonian approach developed here is strictly invariant with respect to choice of phases of the adiabatic electronic eigenstates supersedes the approximate discussion of gauge invariance given earlier by Romero-Rochin and Cina [25] (see also [40]). 

In this second formula, neither the gauge functions nor the spatially dependent coefficients are normally the same as those appearing in the first representation of T. Furthermore, the spatial variable x will frequently be scaled differently from its nondimensionalization in the portion of the domain where... 

Unlike the case of a regular perturbation expansion, we cannot know the gauge functions a priori, and these must be determined as a part of the solution. To obtain governing equations for the unknown concentration functions co and c, we substitute (4-173) into the governing equation, (4-158) (remembering that Da = e ). Now, recalling from Section B that... 

Here the gauge function for the first term must be independent of s because C = 1 at the boundary 7=0. Note again that asymptotic convergence requires that... 

We note that the gauge function for the first term in these expansions has been immediately set equal to unity because the boundary conditions require that the magnitude of the velocity u v be unity (i.e., independent of the small parameter ). 

However, given this information on the gauge function < i, we can now write the governing equation for /i(andySi) by substituting (5-214) into (5-186). The result for the... 

Unlike the regular perturbation expansion discussed earlier, the method of matched asymptotic expansions often leads to a sequence of gauge functions that contain terms like Pe2 In Pe or Pe3 In Pe that are intermediate to simple powers of Pe. Thus, unlike the regular perturbation case, for which the form of the sequence of gauge functions can be anticipated in advance, this is not generally possible when the asymptotic limit is singular In the latter case, the sequence of gauge functions must be determined as a part of the matched asymptotic-solution procedure. 

In the so-called inner region nearest the sphere, the sphere radius is appropriate as the characteristic length scale, and thus equation (9-75) is applicable. To solve this equation for Re <asymptotic expansion, but unlike for (9-77), we do not presuppose any knowledge of the gauge functions, so that... 

However, rather than simply accepting Stokes solution as the first approximation for Re <inner region, we will show how it is obtained in the present framework of matched asymptotic expansions. To do this, we note that the governing equation for fo in the expansion (9-93) is simply the Stokes equation for any choice of the gauge function fo (Re), namely,... 

It is evident that A = 1. There is a mismatch in p of 0(1) for p . However, this mismatch need not concern us at the present level of approximation. A corresponding term, independent of r (because T jl fn is independent of p), will be generated only by a term in the inner expansion (9-164) that has a gauge function 0(Pel/2). At the present leading order of approximation, such terms have not yet been considered. 

Now, with A determined, our solution for the first term in the outer expansion is completed, and we can turn to the problem of obtaining a second term in the asymptotic expansion for 6 in the inner region. In view of the fact that the mismatch between the first terms in the inner and outer expansions has been shown in the previous paragraph to be 0(Pe1/2), it is clear that the gauge function for the second term in the inner expansion (9-164) must be... 

The development status of process control instrumentation lags that of the quality control instruments significantly Nuclear density gauges function in the coal preparation plant environment The slurry concentration meter has application in the intermediate and fine sized coal cleaning circuits and needs to be tested in a preparation plant Other devices, such as ash monitors to control the operation of heavy media baths or jigs are not available and instruments developed for other process industries are not suitable for use in coal preparation plants Modeling studies of the various unit operations are required in order to ascertain the fundamental parameters required to automate the control of these systems Primary process control instrument needs include ash, sulfur, and moisture monitors secondary needs include an on-line washability and ash fusion measurement ... 

Over the years, users of perturbation methods have evolved a shorthand language to express ideas. This reduces repetition and allows compact illustration. We first present the gauge functions, which are used to compare the size of functions, and then we present the order concept, which is convenient in expressing the order of a function (i.e., the speed it moves when e tends small). Finally, we discuss asymptotic expansions and sequences, and the sources of nonuniformity, which cause the solution for 0 to behave differently from the base case. 

In some cases, the following gauge functions are useful (mostly in fluid flow problems)... 

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