Maclaurin

Maclaurin s Integral Test. Suppose X is a series of positive terms and/is a continuous decreasing function such that/(x) > 0 for... 

Swalin then uses a Maclaurin expansion of the Anhenius term to obtain the energy of formation of a cavity in the liquid which permits a small jump, to obtain the equation... 

We have shown that the number of linearly independent invariants of degree k under the permutation group H is equal to the coefficient of in the Maclaurin expansion of (1.27). This represents an important special case of a proposition by Th. Molien. ... 

The form of f 0) can only be determined by experiment, or by means of atomistic hypotheses (cf. Chap. XVIII.). Since ce is known to change continuously and only comparatively slowly with rise of temperature, w e may expand J 0) in a Maclaurin s series (cf. H. M., 95) ... 

In Section 2.4 we have studied the behavior of the gravitational field of the spheroid outside of masses. Now let us focus our attention on the field of attraction inside masses. It may be proper to notice that the determination of the field caused by masses in the spheroid and, in general, by an ellipsoid, was a subject of classical works performed by Maclaurin, Lagrange, Laplace, Poisson, and others. As is well known, the equation of the ellipsoid, when the major axes are directed along coordinate lines is... 

Maclaurin, Colin 34n Mathieu, Brafle Leonard 114n Maxwell, James Clark 39n Mills, Ian In, 352 Milne. William E. 345n... 

In general, the donor and receiver concentrations are exponential functions of time. It is only within the early time period when no more than 10-15% of CD(0) has been transported that the kinetics are essentially linear hence, the tmncated Maclaurin s expansion of Eqs. (5) and (7) leads to the linear relationships... 

Maclaurin s Integral Test. Suppose Is, is a series of positive terms and/is a continuous decreasing function such that/(.r ) > 0 for 1 < x < so and/(n) = an. Then the series and the improper integral j fix) dx either both converge or both diverge. 

Hall, G.E.M., Vaive, J.E., MacLaurin, A. 1996. Selective leaching of the labile organic component of humus and soils with sodium pyrophosphate solution. In EXTECH I a multidisciplinary approach to massive sulphide research in the Rusty Lake-Snow Lake greenstone belts, Manitoba). Geological Survey of Canada Bulletin, 201-213. 

MacLaurin, P. Andrew, K. N. Worsfold, P. J. Flow Injection Analysis. In Process Analytical Chemistry, McLennan, F. Kowalski, B. R. Eds. Blackie Academic and Professional Glasgow, 1995 p. 159. 

This form has the advantage of not containing concentration value m and thus permits the explicit expression of x. from the basic equation. The expansion of the general function given by "Equation 17 into the MacLaurin series with respects to molalities m yields the equation ... 

Typically the experimental measurements are most accurate at small q. Inserting the MacLaurin expansion, Eq. (5.40), of IIo(p) into Eq. (5.63) leads to the MacLaurin expansion [167,187] for the isotropic Compton profile ... 

Hartree-Fock calculations of the three leading coefficients in the MacLaurin expansion, Eq. (5.40), have been made [187,232] for all atoms in the periodic table. The calculations [187] showed that 93% of rio(O) comes from the outermost s orbital, and that IIo(O) behaves as a measure of atomic size. Similarly, 95% of IIq(O) comes from the outermost s and p orbitals. The sign of IIq(O) depends on the relative number of electrons in the outermost s and p orbitals, which make negative and positive contributions, respectively. Clearly, the coefficients of the MacLaurin expansion are excellent probes of the valence orbitals. The curvature riQ(O) is a surprisingly powerful predictor of the global behavior of IIo(p). A positive IIq(O) indicates a type 11 momentum density, whereas a negative rio(O) indicates that IIo(O) is of either type 1 or 111 [187,230]. MacDougall has speculated on the connection between IIq(O) and superconductivity [233]. 

Calculations of IIq(O) are very sensitive to the basis set. The venerable Clementi-Roetti wavefunctions [234], often considered to be of Hartree-Fock quality, get the sign of IIq(O) wrong for the sihcon atom. Purely numerical, basis-set-free, calculations [232,235] have been performed to establish Hartree-Fock limits for the MacLaurin expansion coefficients of IIo(p). The effects of electron correlation on IIo(O), and in a few cases IIq(O), have been examined for the helium atom [236], the hydride anion [236], the isoelectronic series of the lithium [237], beryllium [238], and neon [239] atoms, the second-period atoms from boron to fluorine [127], the atoms from helium to neon [240], and the neon and argon atoms [241]. Electron correlation has only moderate effects on IIo(O). 

Choosing a coordinate system that diagonalizes the Hessian matrix, the MacLaurin expansion of the three-dimensional momentum density II( p) can be written as [241]... 

Values of the MacLaurin coefficients computed from good, self-consistent-field wavefunctions have been reported [355] for 125 linear molecules and molecular ions. Only type I and II momentum densities were found for these molecules, and they corresponded to negative and positive values of IIq(O), respectively. An analysis in terms of molecular orbital contributions was made, and periodic trends were examined [355]. The qualitative results of that work [355] are correct but recent, purely numerical, Hartree-Fock calculations [356] for 78 diatomic molecules have demonstrated that the highly regarded wavefunctions of Cade, Huo, and Wahl [357-359] are not accurate for IIo(O) and especially IIo(O). These problems can be traced to a lack of sufficiently diffuse functions in their large basis sets of Slater-type functions. 

Benko FA, MacLaurin CL, Koffyberg FP (1982). CUWO4 and CusWOe as anodes for the photoelectrolysis of water. Mater Res Bull 17 133-136... 

In the following, we show that the coefficients a , in Eq. (3.31) are related to the derivatives of the sample wavefunction i ) with respect to X, y, and z at the nucleus of the apex atom in an extremely simple way. (To simplify the notation, we take the nucleus of the apex atom as the origin of the coordinate system, i.e., xo = 0, yo = 0, and zo - 0.) This is similar to the well-known case that the expansion coefficients for a power series are simply related to the derivatives of the function at the point of expansion, the so-called Taylor series or MacLaurin series. We will then obtain the derivative rule again, from a completely different point of view. 

A common situation in thermodynamics is that some property z(x) and its lower derivatives (zf, z", z ",. ..) have been measured at a certain point x0, and one wishes to use this information to approximate the behavior of the function z(x0 + Ax) in the Ax-neighborhood of x0. For this purpose, the fundamental Taylor series (or MacLaurin series, the special case for xo = 0) yields approximations that are useful for sufficiently small Ax ... 

We will denote the solvent by the subscript 0. Usually In t/ is expressed as a Maclaurin series in which only the first term is retained—i.e.,... 

This is an expression of the fact that at constant heat flux the melt thickness initially grows linearly with time. For small times an exact solution of this problem has been given by Evans et al. (E3), who expanded Ai (tji) in a Maclaurin series and found the first five coefficients by direct substitution ... 

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