Third-order estimates

The open circles slightly outside TZ in Fig. 2 are third-order estimates of boundary points obtained by the lower bound method and are accurate to three figures. These results are for three spin-up, spin-down pairs of electrons wandering on a ring with six lattice sites. 

As the number of lattice sites increases, the electrons experience additional correlations, so the representable region shrinks. That is, if TZi is the representable region for a lattice with i = A sites, then TZ4 dTZ dTZ% TZw d . This phenomenon is accurately tracked by the third-order estimates, and Fig. 3 shows that convergence to the limiting case where A oo is rapid. 

Fig.3.1. Estimates of the logarithmic-derivative function for free s electrons compared with the exact result D(x) = x cot x - 1, x = S/E explained in Sect. 4.4. The curve labelled to is the second-order estimate (3.51), E(D) is the third-order estimate (3.50). while Lau is the Laurent expansion (3.30) valid to third order in (E - EV)S2. The potential parameters used in the three estimates are derived in Sect.4.4 and listed in Table 4.4. The two open circles in the figure refer to the points (EVS2,DV) and (EVS2,D ), where EVS2 is K2S2 of Table 4.4...

Fig.4.4. Estimates of the 3d potential function p(E) for chromium. The heavy line is the third-order estimate obtained from (2.9,3.50), the thin line is the second-order estimate obtained from (2.9,3.51), and the broken line is the linear approximation (3.41)...

To estimate the third-order derivatives of the function w with respect to y, we make use of the following fact (see Duvaut, Lions, 1972). Let O d E be a bounded domain with smooth boundary and let u be a distribution on O such that u, Du G Then u G L 0) and there is a constant c,... 

This is the Verlet algorithm for solving Newton s equation numerically. Notice that the term involving the change in acceleration (b) disappears, i.e. the equation is correct to third order in At. At the initial point the previous positions are not available, but may be estimated from a first-order approximation of eq. (16.29). 

The order of the desorption process is estimated in the first place from the shape of the desorption peak, preferably in the l/T scale. The first-order peaks here are clearly asymmetric, the falling branch being steeper than the ascending one. The second-order peaks are near symmetric and are broader. The third-order peaks are even broader and are again asymmetric, but in this case the ascending branch is steeper than the falling one. 

The reaction between Fe(IlI) and Sn(Il) in dilute perchloric acid in the presence of chloride ions is first-order in Fe(lll) concentration . The order is maintained when bromide or iodide is present. The kinetic data seem to point to a fourth-order dependence on chloride ion. A minimum of three Cl ions in the activated complex seems necessary for the reaction to proceed at a measurable rate. Bromide and iodide show third-order dependences. The reaction is retarded by Sn(II) (first-order dependence) due to removal of halide ions from solution by complex formation. Estimates are given for the formation constants of the monochloro and monobromo Sn(II) complexes. In terms of catalytic power 1 > Br > Cl and this is also the order of decreasing ease of oxidation of the halide ion by Fe(IlI). However, the state of complexing of Sn(ll)and Fe(III)is given by Cl > Br > I". Apparently, electrostatic effects are not effective in deciding the rate. For the case of chloride ions, the chief activated complex is likely to have the composition (FeSnC ). The kinetic data cannot resolve the way in which the Cl ions are distributed between Fe(IlI) and Sn(ll). 

Five large basis sets have been employed in the present study of benzene basis set 1, which has been taken from Sadlej s tables [37], is a ( ()s6pAdl6sAp) contracted to 5s >p2dl >s2p and contains 210 CGTOs. It has been previously adopted by us in a near Hartree-Fock calculation of electric dipole polarizability of benzene molecule [38]. According to our experience, Sadlej s basis sets [37] provide accurate estimates of first-, second-, and third-order electric properties of large molecules [39]. 

Various compilations of densities for organic compounds have been published. The early Landolt-Bomstein compilation [23-ano] contained recommended values at specific temperatures. International Critical Tables [28-ano-l] provided recommended densities at 0 °C and values of constants for either a second or third order polynomial equation to represent densities as a function of temperature. This compilation also gave the range of validity of the equation and the limits of uncertainty, references used in the evaluation and those not considered. This compilation is one of the most comprehensive ever published. Timmermans [50-tim, 65-tim], Dreisbach [55-die, 59-die, 61-dre] and Landolt-Bomstein [71-ano] published additional compilations, primarily of experimental data. These compilations contained experimental data along with reference sources but no estimates of uncertainty for the data nor recommended values. 

Third-order susceptibilities of the PAV cast films were evaluated with the third-harmonic generation (THG) measurement [31,32]. The THG measurement was carried out at fundamental wavelength of 1064 nm and between 1500 nm and 2100 nm using difference-frequency generation combined with a Q-switched Nd YAG laser and a tunable dye laser. From the ratio of third-harmonic intensities I3m from the PAV films and a fused quartz plate ( 1 thick) as a standard, the value of x(3) was estimated according to the following equation derived by Kajzar et al. [33] ... 

If a single reaction order must be selected, an examination of the 95 % confidence intervals (not shown) indicates that the two-thirds order is a reasonable choice. For this order, however, estimates of the forward rate constants deviate somewhat from an Arrhenius relationship. Finally, some trend of the residuals (Section IV) of the transformed dependent variable with time exists for this reaction order. 

Some protection against the effect of biases in the estimation of the first-order coefficients can be obtained by running a resolution IV fractional factorial design. With such a design the two-factor interactions are aliased with other two-factor interactions and so would not bias the estimation of the first-order coefficients. In fact the main effects are aliased with three-factor interactions in a resolution IV design and so the first-order effects would be biased if there were third-order coefficients of the form xxx, in... 

The third-order rate constartt fej depends on the simultaneous collision of three reactants and has different units (M" sec" ) from the second-order rate of biological inactivation estimated from Fig. 6. As described below, Reaction 4 actually occurs in two separate steps. 

Second, if each run is performed only once, the effect standard deviation can still be estimated because high-order effects should be zero. A non-zero estimate of a third-order effect, therefore, may be attributed to random error and used to estimate the standard deviation of all effects. If m high-order effects can be calculated, the standard deviation of the effect is estimated as... 

The rate data for individual runs can be used to derive independent estimates of k7 + kg, and these are shown in Table II. Both rate constants for an assumed second-order and third-order rate law are shown. The second-order rate constants show the smaller deviation from constancy, but the total change in concentration of the reactants is relatively small so that the order cannot be definitely proved at present. 

Show how to estimate a polynomial distributed lag model with lags of 6 periods and a third order polynomial using restricted least squares. 

Forsyth153 generated CH3 radicals by heating diethyl ether to 800°C. The hot gas flowed down a tube, and NO was added. The change in CH3 radical concentration was monitored by measuring the rate of metal-mirror removal downstream. The data were analyzed in the same manner as in the later work of Durham and Steacie,127 and k7 was estimated to be 1.2 x 107 M-1 sec-1. However, under Forsyth s conditions, the pressure was about 0.6 torr, so that the reaction must be third order as pointed out by Hoare and Walsh.204 The computed rate constant becomes 1.3 x 1012 M-2 sec-1, which must be an upper limit as wall stabilization may also play some role. 

A kinetic analysis of the results, based on (17) and its O+OH analog, is in satisfactory agreement with observations on a wide variety of flames. These flames are relatively cool, and the concentrations of H and OH exceed their equilibrium values even in the burned gases, so that the observed sodium emission is definitely chemiluminescent. The third order rate coefficients for excitation by H+H and H+OH are estimated to be 8 x 109 and 2x 1010 l2.mole-2.sec-1, corresponding to an efficiency near unity per triple collision. The possible importance of mechanisms of the type (14,15) has not been carefully studied. 

The 95% confidence intervals of the MOS lie in the range of 0.1-0.4. For some items, which differ significantly from the fitted curve, the confidence intervals are given. The correlation and standard error of the estimate (R3=0.9 1 and S3=0.48) are derived from the third order regression line that is drawn using a NAG curve fitting routine. 

The correlation and standard error of the estimate (R3=0.81 and S3=0.35) are derived from the third order regression line that is drawn using a NAG curve fitting routine. 

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