Second-order Taylor series expansion

Now, as in the case of the energy, up to this point, we have worked with the nonsmooth expression for the electronic density. However, in order to incorporate the second-order effects associated with the charge transfer processes, one can make use of a smooth quadratic interpolation. That is, with the two definitions given in Equations 2.23 and 2.24, the electronic density change Ap(r) due to the electron transfer AN, when the external potential v(r) is kept fixed, may be approximated through a second-order Taylor series expansion of the electronic density as a function of the number of electrons,... 

The function U = U(ft) can be approximated by the second-order Taylor series expansion at the point i.e.,... 

The FOCE method uses a first-order Taylor series expansion around the conditional estimates of the t] values. This means that for each iteration step where population estimates are obtained the respective individual parameter estimates are obtained by the FOCE estimation method. Thus, this method involves minimizations within each minimization step. The interaction option available in FOCE considers the dependency of the residual variability on the interindividual variability. The Laplacian estimation method is similar to the FOCE estimation method but uses a second-order Taylor series expansion around the conditional estimates of the 77 values. This method is especially useful when a high degree of nonlinearity occurs in the model [10]. 

Furthermore, stresses were calculated as functions of strain and temperature. For each temperature, each component of stress was fit to a second order Taylor series expansion in terms of the strains, about the P = 1 atm reference volume Vq(T) at each specific temperature. Based on the stresses, the elastic moduli Cij and the Griineisen coefficients ji (i = 1,2,3) of the non-crystalline interlamellar phase were calculated using... 

In this section, the URP (updated reference-point) method proposed originally for stochastic input is explained (Fujita and Takewaki 2011a). This method can be used as an efficient uncertainty analysis to obtain the robustness function a explained in the previous section. Since the URP method takes full advantage of an approximation of first- and second-order Taylor series expansion in the interval analysis, the formulation of Taylor series expansion in the interval analysis and the achievement of second-order Taylor series expansion proposed by Chen et al. (2009) are explained briefly. 

As a simple approximation, an objective function f using second-order Taylor series expansion with only diagonal elements can be rewritten as... 

From Eq. 7, the increment of the objective function can be evaluated by using first- and second-order Taylor series expansion as the sum of the increments of the objective function in each one-dimensional domain. The perturbation Afi(X) of the objective function by the variation of the -th interval parameter X, can be described as... 

Estimation of the Variation of the Objective Function by Second-Order Taylor Series Expansion... 

Fujita K, Takewaki I (2011a) An efficient methodology for robustness evaluation by advanced interval analysis using updated second-order Taylor series expansion. Eng Struct 33(12) 3299-3310... 

The first tenn represents the change in / due to changes in the independent variable x, and the second term representce changes due to the independent variable y. Note that Eq. (1.7) is just a generalization of a first order Taylor series expansion to a function of two variables. 

The evaluation of F allows a third order Taylor series expansion of the path. However, a better approach is to combine the third order Taylor series information with the LQA method. This can be done by noting that the expansion coefficient cecond order expansion coefficients can be split into two parts, one that depends only on second energy derivatives and is included in the LQA method and the other that contains terms that depend on the third energy derivatives. 

Eq. 6.2.6 was solved analytically to obtain the operation curve of the reactor (X vs t). Lumped kinetic parameters were determined by non-linear regression of experimental data using the numerical method of Newton-Raphson with first-order Taylor series expansion. Lumped parameters were smooth functions of temperature all parameters were adequately fitted to second order polynomials except for D that required a fourth order polynomial. The model can be used for reactor temperature optimization and can be extended to prolonged sequential batch operation provided that a sound model for enzyme inactivation is validated (Illanes et al. 2005b). 

Probabilistic response analysis consists of computing the probabilistic characterization of the response of a specific structure, given as input the probabilistic characterization of material, geometric and loading parameters. An approximate method of probabilistic response analysis is the mean-centred First-Order Second-Moment (FOSM) method, in which mean values (first-order statistical moments), variances and covariances (second-order statistical moments) of the response quantities of interest are estimated by using a mean-centred, first-order Taylor series expansion of the response quantities in terms of the random/uncertain model parameters. Thus, this method requires only the knowledge of the first- and second-order statistical moments of the random parameters. It is noteworthy that often statistical information about the random parameters is limited to first and second moments and therefore probabilistic response analysis methods more advanced than FOSM analysis cannot be fully exploited. 

Recall that equations 9.86 and 9.100 have been both derived using only the first-order terms in the Taylor series expansion of our basic kinetic equation (equation 9.77). It is easy to show that if instead all terms through second-order in 6x and 6t are retained, the continuity equation ( 9.86) remains invariant but the momentum equation ( 9.100) requires correction terms [wolf86c]. The LHS of equation 9.100, to second order in (ia (5 << 1, is given by... 

Assuming Taylor series expansion using only zero- and first-order terms (dropping second and higher order terms), we arrive at the linear or linearized system described by... 

In general, if k is the number of factors being investigated, the full second-order polynomial model contains V2 k -t- 1)(A -h 2) parameters. A rationalization for the widespread use of full second-order polynomial models is that they represent a truncated Taylor series expansion of any continuous function, and such models would therefore be expected to provide a reasonably good approximation of the true response surface over a local region of experiment space. 

Functional Taylor series expansion of the functional minimized in Eq. (87), in powers of noK ") = [nGs( ) - gs( )] has been employed first, and Eq. (88) used in the last step. So E " is close to KS correlation energy functional taken for the GS density of HF approximation, corrected by the (much smaller) HF correlation energy, and a small remainder of the second order in the density difference. The last quantity gives an estimate to the large parentheses term of Eq. (28) in [12]. 

We can use exactly the same idea for the integral in the numerator of Eq. (6.10). The Taylor series expansion for the energy expanded around the transition state is, to second order,... 

Using Taylor series expansion, find the forward second-order accurate finite difference expansion for the first derivative of the... 

Keeping this Taylor series expansion truncated at the second order, is equivalent to assuming that the only information available to us is expressed by... 

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