Three-term series

Hunter, I. M. L. (1957). The solving of the three-term series problems. British Journal of Psychology, 48, 286-298. 

Johnson-Laird, P. N. (1982). The three-term series problem. Cognition, 1, 57-82. 

O Connor, N., and Hermelin, B. M. (1972). The re-ordering of three term series problems by blind and sighted children. British Journal of Psychology, 63 (3), 381 — 386. 

Tanford (2,9,10) has considered explicitly the dependence on s of the quantity Xi - x , which is supposed to be the sum of a hydrophobic and an electrostatic contribution. The hydrophobic part is represented as a linear function of the area per chain, which decreases with increasing micelle size r. The electrostatic part depends on the area per head group on the micelle surface it is given as a three-term series of negative powers of this quantity. 

According to what has been said, when ir is expressed through A/ii (Equation 1.2-27), ln(l — V2) in Equation 50 should be approximated by a three-term series expansion... 

Application of an infinite series to practical calculations is, of course, impossible, and truncations of the virial equations are in fact employed. The degree of truncation is conditioned not only by the temperature and pressure but also by the availability of correlations or data for the virial coefficients. Values can usually be found for B (see Sec. 2), and often for C (see, e.g., De Santis and Grande, ATChP J., 25, pp. 931-938 [1979]), but rarely for higher-order coefficients. Application of the virial equations is therefore usually restricted to two- or three-term truncations. For pressures up to several bars, the two-term expansion in pressure, with B given by Eq. (4-188), is usually preferred ... 

Molecules that are composed of atoms having a maximum valency of 4 (as essentially all organic molecules) are with a few exceptions found to have rotational profiles showing at most three minima. The first three terms in the Fourier series eq. (2.9) are sufficient for qualitatively reproducing such profiles. Force fields which are aimed at large systems often limit the Fourier series to only one term, depending on the bond type (e.g. single bonds only have cos (3u ) and double bonds only cos (2u))). 

Fig. 12 presents the variation of the mesophase moduli, Ej(r), for the various inclusion-volume fractions, versus the extent of the mesophase Ar, normalized to the highest inclusion-volume fraction of 25 percent. This was done in order to show the similarity of variation of the Ermodulus for the various values of uf for this series of composites, possessing the same adhesion properties between them. These normalized patterns are equivalent with those presented in Fig. 11 for the three-term unfolding model, since the differences between corresponding values of the two versions of the model are insignificant. 

Fig. 15. The variation of the adhesion coefficient A = (ri, — t 2) for the three-term unfolding model and the exponent 2r for the two-term model of a series of E-glass-epoxy fiber composites, versus the fiber-volume content uf...

The general solution of the Eq. (50) can be obtained in power series form. Under the condition that Res < 0(1), (r ) can be approximated by only including the first three terms in the power series with good accuracy ... 

To construct an image including the lowest nontrivial Fourier components, only three terms in the Fourier series are significant. Those are Go(z), G i(z), and Gdz)- Because of the reflection symmetry of the conductance function g(x,z), the last two Fourier coefficients are equal, and are denoted as Gi(z). Up to this term. 

If we expand Eq. (5) in a power series in the magnetic field about = 0 the zeroth order term will be zero and the three terms that are linear in will be sufficient to describe many MCD spectra. We will therefore consider the first derivative of Eq. (5) with respect to B. An important exception where terms of higher order in the magnetic field are often needed will be described in Section II.A.5. 

The first three terms represent the forward Euler algorithm operating on the exact solution, with the last term [in square brackets] providing a measure of the local truncation error. The local truncation error can be identified through a Taylor series expansion of the solution about the time tn ... 

The first three terms represent the implicit Euler algorithm and the remaining [bracketed] term represents the local truncation error. A Taylor series expansion about tn+ (in the negative t direction) yields an expression for y(t )... 

The inclusion of only three terms may be insufficient, since in the related molecule butadiene four terms in the Fourier series are found to be necessary to describe appropriately the observed frequencies172). 

Definition of three terms that are commonly used interchangeably but nonetheless have distinct meanings will aid our discussion on the controls of soil C dynamics. In the most general sense, a process is a series of steps leading to a result in the... 

This behaviour has been studied by Barbarella et al.,47,87 who have reported the chemical shifts for a series of cyclic and acyclic sulphides, sulphoxides and sulphones. They have found that there is a proportionality between the 33S chemical shift and the sulphur atomic charge only for thiirane derivatives. According to the authors, this behaviour can be explained if one considers the complex dependence of the paramagnetic contribution to the shielding constant, <7p, on the three terms AE, (r-3) and bond order character (Equation (2),... 

The first three terms of the series are overestimating the available volume, since they do not account for the overlapping of the forbidden regions for triplets, and a new correction (proportional to the third power of density) should be included, and so on. The calculation of the remaining terms of the (slowly) converging series is however increasingly difficult. The first two terms of the... 

To seek a reasonable accurate analytical approximation for the available area, as a function of 6S = Nsnr /A and 6y = Nynr /A one should have accurate values for a reasonable number of coefficients in the low-density expansion of the binaiy RSA model, which is not a trivial task. Even for binaiy mixtures of disks at equilibrium, a problem that received much more attention than RSA, analytical expressions are known only for the first three terms of the virial expansion [21], The values of the fourth and fifth terms, obtained using laborious numerical calculations, were reported only for a few values of y and molar fractions of the two types of disks [22], In the non-equilibrium RSA of binaiy particles, one should take into account, when calculating the higher terms of the series, not only various y and molar fractions, but also the order of deposition of particles. Furthermore, as already noted, it is not clear whether the involved calculations needed to obtain the next unknown terms of the low-density expansion would improve much the accuracy of estimating the jamming coverage. 

For small displacements, of the order of vibrational amplitudes at room temperature, the terms in the power series expansion (1) converge fairly rapidly, and higher-order terms are related to successively smaller-order effects in the spectrum, so that they become more and more difficult to determine. Almost all calculations to this date have been restricted to determining quadratic, cubic, and quartic force constants only [the first three terms in equation (1)], and in this Report we shall not consider higher-than-quartic terms in the force field. The paper by Cihla and Chedin11 is one of the few exceptions in which force constants involving up to the sixth power have been determined for a polyatomic molecule, namely COa. 

The two forms of the virial expansion given by Eqs. (3.10) and (3.11) are infinite series. For engineering purposes their use is practical only where convergence is very rapid, that is, where no more than two or three terms are required to yield reasonably close approximations to the values of the series. This is realized for gases and vapors at low to moderate pressures. 

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