Linear Taylor expansion

The mass action variable X can be rendered into an experimentally more meaningful form. To do this, let X be a reference value for given values of the concentration and temperature T. Then, by a linear Taylor expansion of the dimensionless free energy f3g around the reference temperature T, we have... 

In a real reactor the variation of the temperature over the bed must be between those predicted by the two above equations. Equations 5.52 - 5.55 and 5.57 can be used to check whether ideal mixing conditions are achieved or not. For practical purposes these equations can be simplified because only small concentration and temperature variations are considered. In this case the inlet reaction rate can be approximated by a linear Taylor expansion ... 

Any model linear in its location parameters 0 has a uniform Jeffreys prior p 6i) over the permitted range of 0 . This condition occurred in Examples 5.6 and 5.8, where the model = p + eu with location parameter p was used. The Jeffreys prior p(0 ) is likewise uniform for any model nonlinear in 0 , over the useful range of its linearized Taylor expansion that we provide in Chapters 6 and 7. 

Step 2 Chck the = to solve the problem. This time the program solves the problem for a variety of a. In fact, it finds the solution and its derivative with respect to a each time. Then the program uses a linear Taylor expansion, Eq. (9.29), to provide a good starting guess for the iterative solution at the new a ... 

The free energy derivatives are also related to the coefficients in a Taylor expansion of the free energy with respect to X. In the case of linear coupling, we let Eba = XdJ-a — Up,)lkT in Eq. (9) we obtain... 

Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

Determination of confidence limits for non-linear models is much more complex. Linearization of non-linear models by Taylor expansion and application of linear theory to the truncated series is usually utilized. The approximate measure of uncertainty in parameter estimates are the confidence limits as defined above for linear models. They are not rigorously valid but they provide some idea about reliability of estimates. The joint confidence region for non-linear models is exactly given by Eqn. (B-34). Contrary to ellipsoidal contours for linear models it is generally banana-shaped. 

The simplest method for propagation of the analytical errors into the °Th/U age equation (Eqn. 1) involves linear expansion (Albarede 1995, ch. 4.3) of the effect on the calculated age of very small perturbations of the measured ratios (in effect, a Taylor expansion using only the first term). As long as the effect of the errors of the measured ratios on the age is not a large fraction of the age itself, this method will yield acceptably accurate age-errors with minimal effort. 

Since x, and m, are functions of time, we need to linearize (10-26). A first order Taylor expansion of x is... 

Taylor expansion of (6.89) is not based on a physical model, and more-complicated nonlinear forms could be used. However, nonlinear extrapolation methods seem to offer little improvement over their linear counterparts [58]. Larger improvements appear to be achieved using the cumulative integral extrapolation method discussed in Sect6.7.3. 

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 ... 

The authors describe the use of a Taylor expansion to negate the second and the higher order terms under specific mathematical conditions in order to make any function (i.e., our regression model) first-order (or linear). They introduce the use of the Jacobian matrix for solving nonlinear regression problems and describe the matrix mathematics in some detail (pp. 178-181). 

The method developed for linear constraints is extended to nonlinearly constrained problems. It is based on the idea that the nonlinear constraints linear Taylor series expansion around an estimation of the solution (xi, ut). In general, measurement values are used as initial estimations for the measured process variables. The following linear system of equations is obtained ... 

As with almost any other non-linear problem that has to be solved iteratively, linearisation via a Taylor expansion with truncation after very few elements, is the solution. 

As done previously, in The Newton-Raphson Algorithm (p.48), we neglect all but the first two terms in the expansion. This leaves us with an approximation that is not very accurate but, since it is a linear equation, is easy to deal with. Algorithms that include additional higher terms in the Taylor expansion, often result in fewer iterations but require longer computation times due to the calculation of higher order derivatives. 

In the linearization method, the nonlinear model of Eq. (40) is linearized by a truncated Taylor expansion ... 

The general approach used with nonlinear models, such as Eq. (40) is to linearize by a Taylor expansion [Eq. (41)] and apply the linear theory of Section III,C,1. 

In nonequilibrium steady states, the mean currents crossing the system depend on the nonequilibrium constraints given by the affinities or thermodynamic forces which vanish at equihbrium. Accordingly, the mean currents can be expanded in powers of the affinities around the equilibrium state. Many nonequilibrium processes are in the linear regime studied since Onsager classical work [7]. However, chemical reactions are known to involve the nonlinear regime. This is also the case for nanosystems such as the molecular motors as recently shown [66]. In the nonlinear regime, the mean currents depend on powers of the affinities so that it is necessary to consider the full Taylor expansion of the currents on the affinities ... 

The quantum alternative for the description of the vibrational degrees of freedom has been commented by Westlund et al. (85). The comments indicate that, to get a reasonable description of the field-dependent electron spin relaxation caused by the quantum vibrations, one needs to consider the first as well as the second order coupling between the spin and the vibrational modes in the ZFS interaction, and to take into account the lifetime of a vibrational state, Tw, as well as the time constant,T2V, associated with a width of vibrational transitions. A model of nuclear spin relaxation, including the electron spin subsystem coupled to a quantum vibrational bath, has been proposed (7d5). The contributions of the T2V and Tw vibrational relaxation (associated with the linear and the quadratic term in the Taylor expansion of the ZFS tensor, respectively) to the electron spin relaxation was considered. The description of the electron spin dynamics was included in the calculations of the PRE by the SBM approach, as well as in the framework of the general slow-motion theory, with appropriate modifications. The theoretical predictions were compared once again with the experimental PRE values for the Ni(H20)g complex in aqueous solution. This work can be treated as a quantum-mechanical counterpart of the classical approach presented in the paper by Kruk and Kowalewski (161). 

Remark 1 The right-hand side of the above inequality (2.1) is a linear function in x and represents the first-order Taylor expansion of f(x) around x° using the vector d instead of the gradient vector of /(x) at x°. Hence, d is a subgradient of /(x) at x° if and only if the first-order Taylor approximation always provides an underestimation of /(x) for all x. 

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... 

This equation gives the dynamics of the quantum-classical system in terms of phase space variables (R, P) for the bath and the Wigner transform variables (r,p) for the quantum subsystem. This equation cannot be simulated easily but can be used when a representation in a discrete basis is not appropriate. It is easy to recover a classical description of the entire system by expanding the potential energy terms in a Taylor series to linear order in r. Such classical approximations, in conjunction with quantum equilibrium sampling, are often used to estimate quantum correlation functions and expectation values. Classical evolution in this full Wigner representation is exact for harmonic systems since the Taylor expansion truncates. 

This parametrization can be very demanding, hence one goes a step further and only calculates the electronic structure at equilibrium and the leading term in a Taylor expansion on the nuclear coordinates the electron-vibration interaction is linearized. 

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