Asymptotic relationships

A series of o-cresol novolac-PDMSX materials with a range of silicon contents (3.2-16.1 wt %) were prepared to examine the O2 REE response and to determine whether a solubility limit existed as a function of wt % silicon. All copolymer preparations exhibited good solubility in dilute TMAH and an asymptotic relationship (12) between wt % silicon and film thickness loss (3,12) during an 02 plasma RIE process was found. 

FIGURE 20.4 Asymptotic relationship between the percent conversion, as a step-growth polymerization proceeds, and the polymer molecular weight. 

The transepithelial electrical resistance measures predominantly the permeability of tight junctions for small ionic solutes. A partial disruption of tight junctions would not be detected by changes in permeability but by a drop in transepithelial resistance, which is reflected by an asymptotic relationship of... 

If the ratio of flow rate to mass of wetted soil (L day -1 g"l) is plotted on the abscissa, and silica release rate [silica export per square meter of wetted minerul (mol m-2 s "1)] is plotted on the ordinate, an asymptotic relationship for silica release rate is determined (Fig. 18). Silica release rates reach a maximum of 10 11 to 10-12 at a flow-rate mass ratio greater than 10 3,4 L day 1 g. TliiN corresponds to the flow regime and weathering rates most frequently reported in... 

Finally, it is possible to develop an asymptotic relationship for the numerical incomparability at medium entropy. The minimally diverse partitions (furthest to the left on the diagram lattice) are those that minimize the number of rows (for a given entropy - or vertical position on the lattice). This is accomplished if the rows have nearly the same number of objects (boxes). We let m be the number of rows in a partition in the left-hand chain then there are approximately n/m boxes in each row for partitions on the left-hand chain to minimize m (or number of rows). 

Similarly, for F2 the follo ving asymptotic relationship can be obtained ... 

The asymptotic relationship between pressure and flow is due to the effect of the concentration polarization. At low pressure, low solute concentration of the feed, and high fluid velocity, the effects of the concentration polarization are minimal and the permeate flux is affected by transmembrane pressure. Deviations from the linear relationship between flow and pressnre are observed at high pressures, independent of other operating parameters, due to the consolidation of the polarized gel layer of the solute. 

Let us now investigate the accuracy of the asymptotic relationship shown in equation (12). For power-law potentials, a comparison of equation (12) with numerical calculations is given in ref. [18]. Even for the ground state (/ = 0), the precision of this simple expression is surprisingly good, especially at 3 = 1 and 3 = 4 (the anharmonic oscillator). Note that in this case, the coefficients change by many orders of magnitude ... 

Cicnerally, the relationships between JV and /V, V2 and 1)2 are very soj)histicaled how -ever, be N rather large, b can be chosen large enough to obtain a simple asymptotic relationship b< tween the quantities with and without a tilde. 

Fig. 4.5 shows an example of an exact solution for this situation. It can be proved that the QSS concentration of B can be approximated closely by the following asymptotic relationship ... 

Although the classical theory is not correct in predicting that the observed radiation will consist of frequencies occurring in the motion of the system, there is an asymptotic relationship between the frequencies predicted by the classical and by the quantum theory, known as Bohr s correspondence theorem for frequencies. According to this theorem the frequencies emitted and absorbed by a quantum system approach asymptotically the classical frequencies of the system as the quantum numbers of the initial and final states are increased. The intensities of the quantum transitions will likewise asymptotically approach the intensities calculated classically, as the quantum numbers increase. 

It will be seen from Example 14. that for > 1 the relationship = (n J varies very slowly. From the asymptotic relationship (4.33) follows, that in the case c) the slope of the relationship = ( 2) is proportional to the magnitude of the difference V2 — v. Assuming the stoichiometric coefficients to be mutually non-divisible integers, there applies in Example 14 (v2 — v) = 1. Let us chose an example in which the difference (v2 — v) is considerably greater. 

One may ask then what is the reason for the usefulness of the low-order polarization theory calculations. The answer is that at sufficiently large R the truncated polarization series can provide a very good approximation to the interaction energy. This is a consequence of the asymptotic relationship ... 

The main purpose of this chapter is to demonstrate the equivalence of the ray and asymptotic modal descriptions of multimode waveguides. We begin by deriving the asymptotic relationship between a mode of the waveguide and the equivalent family of... 

Equation 19.5 shows film concentration increasing with increasing growth time, t (s ). For a large growth time, the right side of Equation 19.5 approaches the asymptotic relationship ... 

The SSE has an absolute minimum of 0, associated with a data distribution where all the values fall into a single bin and a maximum of 1.0, where each bin is occupied by an equal number of data counts. As we shall see, SSE is not independent of boundary effects (described later) underlying the data and there is an asymptotic relationship associated with the number of bins, which can be ignored for most practical comparisons. 

In addition to this asymptotic relationship associated with the number of bins used in a given study, the treatment of data outliers affects SSE calculations, just as it would influence the analysis of any histogram. A large body of literature associated with both of these topics exists (see Refs. 19-21). Surprisingly, there is no known optimum value for the number of bins chosen for a histogram, but commonly accepted rules exist. For example, one postulate is that the bin width should be proportional to both the standard deviation of the... 

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