Taylor expansion developments

All expressions can be developed by means of the Taylor expansion. With these, we can express a function close to a given point (here P ) in terms of the weighted sum of a series of derivatives at the point there are forward- and backward-looking series  

If h is small, we can safely neglect all terms from h/2 .. onwards and thus obtain the same expression as Eq. 3.10 but now we have an idea of the error we are making, which is seen to be of the order of h (written 0(h)). 

2) Backward difference scheme dy/dx in Eq. 3.17 is isolated and, in a manner similar to the forward scheme described above, get the same expression as in Eq. 3.11, again with an error of 0(h). 

if we neglect the term in h and higher terms, this comes to the same as Eq. 3.12. The error is O(h ), much smaller than that for the other two schemes above. 

We have now derived our discretisation approximations, Eqs. 3.10 to 3.13, both by intuitive guessing and mathematically. 

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In the next section we derive the Taylor expansion of the coupled cluster cubic response function in its frequency arguments and the equations for the required expansions of the cluster amplitude and Lagrangian multiplier responses. For the experimentally important isotropic averages 7, 7i and yx we give explicit expressions for the A and higher-order coefficients in terms of the coefficients of the Taylor series. In Sec. 4 we present an application of the developed approach to the second hyperpolarizability of the methane molecule. We test the convergence of the hyperpolarizabilities with respect to the order of the expansion and investigate the sensitivity of the coefficients to basis sets and correlation treatment. The results are compared with dispersion coefficients derived by least square fits to experimental hyperpolarizability data or to pointwise calculated hyperpolarizabilities of other ab inito studies. 

We develop y into a Taylor-series around a set of initial values for the k- which must not deviate too much from the optimised final values. With this condition we may truncate the Taylor-expansion after the linear terms and obtain the following system of linear relations ... 

Spectroscopic applications usually require us to go beyond single-point electronic energy calculations or structure optimizations. Scans of the potential energy hypersurface or at least Taylor expansions around stationary points are needed to extract nuclear dynamics information. If spectral intensity information is required, dipole moment or polarizability hypersurfaces [202] have to be developed as well. If multiple relevant minima exist on the potential energy hyper surface, efficient methods to explore them are needed [203, 204],... 

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

A new approach to the application of group theory in the study of the physical properties of crystals, which is more powerful than the direct method described in Section 15.2, has been developed by Nowick and is described fully in his book Crystal Properties via Group Theory (Nowick (1995)). A brief outline of Nowick s method will be given here. The equilibrium physical properties of crystals are described by constitutive relations which are Taylor expansions of some thermodynamic quantity Yt in terms of a set of thermodynamic variables Xj. Usually, only the first term is retained giving the linear relations... 

Consider again Fig. 3.1, and the point at. i 2, expressed as a Taylor series development going from aq. Note that the symbols yz, f x2) and /(aq + h) are all synonymous. The Taylor expansion is... 

It is possible to develop the point at xi going backward from that at x2, again using the Taylor expansion ... 

In the relation (3.231) s represents the number of experimental points located on the Zg coordinate while r characterizes the time position when a measure is executed. The base of the development of the Newton-Gauss gradient technique resides in the Taylor expansion Y(z,t, P) near the starting vector of parameters Po ... 

It is well known that using an exponential or power function can also describe the portion of a polynomial curve. Indeed, these types of functions, which can represent the relationships between the process variables, accept to be developed into a Taylor expansion. This procedure can also be applied to the example of the statistical process modelling given by the general relation (5.3) [5.20]. 

Recently there has been a great deal of interest in nonlinear phenomena, both from a fundamental point of view, and for the development of new nonlinear optical and optoelectronic devices. Even in the optical case, the nonlinearity is usually engendered by a solid or molecular medium whose properties are typically determined by nonlinear response of an interacting many-electron system. To be able to predict these response properties we need an efficient description of exchange and correlation phenomena in many-electron systems which are not necessarily near to equilibrium. The objective of this chapter is to develop the basic formalism of time-dependent nonlinear response within density functional theory, i.e., the calculation of the higher-order terms of the functional Taylor expansion Eq. (143). In the following this will be done explicitly for the second- and third-order terms... 

If the temperature in a nebula is not uniform, Te-based empirical abundances are biased. Peimbert (1967) developed a mathematical formulation to evaluate the bias. It is based on the Taylor expansion of the average temperature... 

One year after INDO/S, the method SINDOl (symmetrically orthogonalized INDO/1) by Nanda and Jug [58] was introduced. Originally developed for organic compounds of first-row elements, it was later extended to elements of the second and third row [59,60]. This method has several distinct features. The most important is that the orthogonalization transformation [Eq. (14)] is taken into account by a Taylor expansion. The matrix S-1/2 is approximated as... 

A central problem in the development of the theory is the formulation of a suitable expression for A[p] for the nonuniform system of interest. In treatments of SFE, two kinds of approach have been pursued. The first approach is to make a functional Taylor expansion of Aex[p] the excess... 

Because of space limitations we considered only macroscopic kinetic systems, but the approach can be also applied to experiments of single-molecule kinetics [10]. The suggested approach is in an early stage of development. An important issue is extending the approach to nonneutral systems, for which the kinetic and transport properties of the labeled species are different from the corresponding properties of the unlabeled species. There are two different regimes (a) If the deviations from neutrality are small, the response can be represented by a functional Taylor expansion the terms of first order in the functional Taylor expansion correspond to the linear response law. (b) For large deviations, a phase linearization approach is more appropriate. 

Another possibility is to adapt to In A a method developed by Clarke and Glew (32) for the representation of In Kp as a function of temperature, where Kp is an equilibrium constant. In this method, AHridH at temperature T) is expressed as a Taylor expansion round its value at some reference temperature 6. 

Equation 1 expresses the fact that the failure domain D is measured by means of probability measure. It is not easy to calculate Pf using Equation 1, therefore many techniques are developed in the literature. The well known approaches are the FORM/SORM (respectively, First Order Reliability Methods and Second Order Reliability Methods) that consists in using a transformation to change variables into an appropriate space where vector U = T X) is a Gaussian vector with uncorrelated components. In this space, the design point, , is determined. Around this point, Taylor expansion of the limit state function is performed at first order or second order respectively for FORM or SORM method (Madsen et al). In the case of FORM, the structure reliability index is calculated as ... 

The asymptotic relations (Equations 7.9 and 7.10) are exact for any one-component liquid with spherical interactions, except when the Taylor expansion (Equation 7.7) fails to hold, which is near a critical point (Fisher 1964). At the critical point, the DCF becomes a nonanalytic function, and the correlations develop a nonexponential algebraic decay in l/r +i where ii = 0.041 is a critical exponent (Fisher 1964). In all that follows, we will stay away from criticality, hence the OZ forms. Equations 7.9 and 7.10, are essentially exact. When the critical point is approached, the isothermal compressibility diverges, and the analysis above shows that the correlation length diverges as the square root of the compressibility. The divergence of leads to a Coulomb decay of the pair correlation in Equation 7.10, but the correct exponent is slightly faster than pure Coulomb decay. 

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