Complete second order

This is, then, the regression sum of squares due to the first-order terms of Eq. (69). Then, we calculate the regression sum of squares using the complete second-order model of Eq. (69). The difference between these two sums of squares is the extra regression sum of squares due to the second-order terms. The residual sum of squares is calculated as before using the second-order model of Eq. (69) the lack-of-fit and pure-error sums of squares are thus the same as in Table IV. The ratio contained in Eq. (68) still tests the adequacy of Eq. (69). Since the ratio of lack-of-fit to pure-error mean squares in Table VII is smaller than the F statistic, there is no evidence of lack of fit hence, the residual mean square can be considered to be an estimate of the experimental error variance. The ratio... 

When incorporated into the complete second-order model, we obtain... 

Trivedi et al. utilized Sorby s experimental data for water-ethanol-propylene glycol and Ltted to a complete second order polynomial model and performed a stepwise regression to arrive at following equation where andy represent fractions of ethanol and propylene glycol, respectively ... 

For this case of second order complete plan, the specificity of the matrix of the coefficients results in an assembly of relations directly giving the regression values of the coefficients. In this example, where the complete second order plan is based on a 2 CFE, these relations are written as follows ... 

N gives the total number of experiments in the plan. When we use a complete second order plan, it is not necessary to have parallel trials to calculate the reproducibility variance, because it is estimated through the experiments carried out at the centre of the experimental plan. The model adequacy also has to be examined with the next procedure ... 

As the reaction mechanism is not known, it is not possible to exclude any two-variable interaction effect from consideration. A complete second order interaction response surface model was therefore attempted ... 

To establish a complete second order interaction model, a Resolution V (or higher) design must be used. Such designs can be constructed by using four-variable interaction columns to define the "extra" variables, e.g, 2 (I = 12345), 2 (I = 12346 = 12357 = 12458 = = 13459 = 2345 10)... 

Shabaev et al. 2000a. for a recent review), the complete second-order calculation is not yet finished. 

Rayleigh and Raman scattering are two-photon processes in which absorption of a photon is followed coherently by emission of the same and of a different photon, respectively. They are thus second-order processes if we use a description in which the interaction between matter and radiation is used as a perturbation. The complete second-order treatment (Sushchinskii, 1972 Behringer, 1974) is outside the scope of this contribution. We shall start with the simplified description of the scattering tensor that is familiar from many previous treatments. The scattering tensor describing the Raman transition from the molecular eigenstate / > to /> is represented by a 3 X 3 matrix with (molecule-fixed) Cartesian components (in atomic units, ft = 1)... 

Abstract. This article reviews from both theoretical and numerical aspects three non-equivalent complete second-order formulations of quantum dissipation theory, in which both the reduced dynamics and the initial canonical thermal equilibrium are properly treated in the weak system-bath coupling limit. Two of these formulations are rather familiar as the time-local and the memory-kernel prescriptions, while another which can be termed as correlated driving-dissipation equations of motion will be shown to have the combined merits of the two conventional formulations. By exploiting the exact solutions to the driven Brownian oscillator system, we demonstrate that the time-local and correlated driving-dissipation equations of motion formulations are usually better than their memory-kernel counterparts, in terms of their applicability to a broad range of system-bath coupling, non-Markovian, and temperature parameters. Numerical algorithms are detailed for an efficient evaluation of both the reduced canonical thermal equilibrium state and the non-Markovian evolution at any temperature, in the presence of arbitrary time-dependent external fields. 

Recently we have constructed a complete second-order QDT (CS-QDT), in which all excessive approximations, except that of weak system-bath interaction, are removed [38]. Besides two forms of CS-QDT corresponding to the memory-kernel COP [Eq. (1.2)] and the time-local POP [Eq. (1.3)] formulations, respectively, we have also constructed a novel CS-QDT that is particularly suitable for studying the effects of correlated non-Markovian dissipation and external time-dependent field driving. This paper constitutes a review of the three nonequivalent CS-QDT formulations [38] from both theoretical and numerical aspects. Concrete comparisons will be carried out in connection with the exact results for driven Brownian oscillator systems, so that sensible comments on various forms of CS-QDT can be reached. Note that QDT shall describe not only the evolution of p(t), but also the reduced thermal equilibrium system as p t oo) = peq(7 )-... 

Obviously, 0 )2 contains not only the complete second-order contributions, but also a partial sum from all higher orders. Moreover, we may obtain the second-order expression for the coordinate response function, [x( )]2 = ([9(0) (0)])25 either via the direct application of linear response theory to the CS-QDT formulation in Sec. 4.2, or, equivalently, from the comparison between Eq. (4.11b) and Eq. (4.6d). 

Although MAS is very widely applied to non-integer spin quadnipolar nuclei to probe atomic-scale structure in solids, such as distinguishing AlO and AlOg enviromnents [21], simple MAS about a single axis caimot produce a completely averaged isotropic spectrum. As the second-order quadnipole interaction contains both... 

Figure B3.6.3. Sketch of the coarse-grained description of a binary blend in contact with a wall, (a) Composition profile at the wall, (b) Effective interaction g(l) between the interface and the wall. The different potentials correspond to complete wettmg, a first-order wetting transition and the non-wet state (from above to below). In case of a second-order transition there is no double-well structure close to the transition, but g(l) exhibits a single minimum which moves to larger distances as the wetting transition temperature is approached from below, (c) Temperature dependence of the thickness / of the enriclnnent layer at the wall. The jump of the layer thickness indicates a first-order wetting transition. In the case of a conthuious transition the layer thickness would diverge continuously upon approaching from below.

Kinetic measurements were performed employii UV-vis spectroscopy (Perkin Elmer "K2, X5 or 12 spectrophotometer) using quartz cuvettes of 1 cm pathlength at 25 0.1 C. Second-order rate constants of the reaction of methyl vinyl ketone (4.8) with cyclopentadiene (4.6) were determined from the pseudo-first-order rate constants obtained by followirg the absorption of 4.6 at 253-260 nm in the presence of an excess of 4.8. Typical concentrations were [4.8] = 18 mM and [4.6] = 0.1 mM. In order to ensure rapid dissolution of 4.6, this compound was added from a stock solution of 5.0 )j1 in 2.00 g of 1-propanol. In order to prevent evaporation of the extremely volatile 4.6, the cuvettes were filled almost completely and sealed carefully. The water used for the experiments with MeReOj was degassed by purging with argon for 0.5 hours prior to the measurements. All rate constants were reproducible to within 3%. 

Assuming complete binding of the dienophile to the micelle and making use of the pseudophase model, an expression can be derived relating the observed pseudo-first-order rate constant koi . to the concentration of surfactant, [S]. Assumirg a negligible contribution of the reaction in the aqueous phase to the overall rate, the second-order rate constant in the micellar pseudophase lq is given by ... 

Although the proportion of nitric acid present as nitronium ions does not change between 90% and 100% sulphuric acid, the rate constants for nitration of most compounds decrease over this rai e. Fig. 2.1 illustrates the variation with acidity of the second-order rate constants of the nitration of a series of compounds of widely differing reactivities. Table 2.4 lists the results for nitration in 95% and 100% acid of a selection of less completely investigated compounds. 

Second-order rate coefficients for nitration in sulphuric acid at 25 °C fall by a factor of about 10 for every 10 % decrease in the concentration of the sulphuric acid ( 2.4.2). Since in sulphuric acid of about 90% concentration nitric acid is completely ionised to nitronium ions, in 68 % sulphuric acid [NO2+] io [HNO3]. The rate equation can be written in two ways, as follows ... 

For a base the stoichiometric second-order rate constant which should be observed, imder conditions where ionisation to the nitronium ion is virtually complete, namely > 90 % H2SO4, if nitration were limited to the free base and occurred at every encounter with a nitronium ion, would be ... 

The rates of reaction of phenacyl bromide with thiosemicarbazide and its phenylated derivative were determined by conductivity measurements in ethanol (517). The reaction is second order up to 85% completion. The activation energies are 10.5 to 11.3 kcal/mole with the phenyl thiosemicarbazide and 8.5 to 9.3 kcal/mole for the unsubstituted derivatives. 

Earlier we noted that a response surface can be described mathematically by an equation relating the response to its factors. If a series of experiments is carried out in which we measure the response for several combinations of factor levels, then linear regression can be used to fit an equation describing the response surface to the data. The calculations for a linear regression when the system is first-order in one factor (a straight line) were described in Chapter 5. A complete mathematical treatment of linear regression for systems that are second-order or that contain more than one factor is beyond the scope of this text. Nevertheless, the computations for... 

Perikinetic flocculation is the first stage of flocculation, induced by the Brownian motion. It is a second-order process that quickly diminishes with time and therefore is largely completed in a few seconds. The higher the initial concentration of the soflds, the faster is the flocculation. 

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