Linear approximation of the

Next let us turn our attention to models described by a set of ordinary differential equations. We are interested in establishing confidence intervals for each of the response variables y, j=l,...,/w at any time t=to. The linear approximation of the output vector at time to,... 

Since both spot price and quantity are modeled as variables, the resulting optimization problem of maximizing turnover is quadratic. In the following, we show how a linear approximation of the turnover function can be achieved (see also Habla 2006). This approach is based on the concavity property of the turnover function and the limited region of sales quantity flexibility to be considered. Approximation parameters are determined in a preprocessing phase based on the sales input and control data. The preprocessing is structured in two phases as shown in table 25 ... 

In summary, for a query point 09 in the neighborhood of 0q IS AT provides a linear approximation of the form... 

Combining equations (5.4.86) to (5.4.88) gives the linear approximation of the matrix ffSx pftT as... 

The basic idea is very simple In many scenarios the construction of an explicit kinetic model of a metabolic pathway is not necessary. For example, as detailed in Section IX, to determine under which conditions a steady state loses its stability, only a local linear approximation of the system at this respective state is needed, that is, we only need to know the eigenvalues of the associated Jacobian matrix. Similar, a large number of other dynamic properties, including control coefficients or time-scale analysis, are accessible solely based on a local linear description of the system. 

This linear affinity approximation does not always correspond to the linear approximation of kinetic polynomial - (Bo)/(Bj). This happens only when degree p of cyclic characteristic in Proposition 1 (see Equation (34)) is one. If p>l, linear approximation of the kinetic polynomial does not correspond to... 

From a linear law or a linear approximation of the exponential law, the magnitude of the mass bias can be inferred from the measurement of this ratio in the sample... 

The basic idea is the same as in the false position method, i.e., local linear approximation of the function. The starting interval [x x2J does not, however, necessarily include the root. Then the straight line through the... 

We consider a spherical particle aggregate with radius r0 surrounded by a concentric boundary layer of thickness 8 (Fig. 19.16). Transport into the aggregate is described by the linear approximation of the radial diffusion model. Thus, the total flux from the particle to the fluid is given by Eq. 19-85 ... 

The consistent derivation and analysis of the Waite-Leibfried equations is presented in Chapter 4. We show that this theory is the linear approximation of the exact many-particle formalism. Its relation to the Smoluchowski theory is also established. 

Then the piecewise linear approximation of the approximation of the concave function C x ) can be formulated as... 

Remark 1 Note that 71,72,73,74 are known points. Hence, the unknown continuous variables are 1, Ax,A2,A3, A4, and the binary variables are yi,y2,3/3. As a result of this three-piece linear approximation of the concave function C(xi) we have increased the number of continuous variables by four, introduced three binary variables, two linear constraints on the continuous variables, four logical constraints and one constraint on the binary variables only. 

A linear approximation of the potential is certainly too sweeping a simplification. In reality, Vr varies with the internuclear separation and usually rises considerably at short distances. This disturbs the perfect (mirror) reflection in such a way that the blue side of the spectrum (E > Ve) is amplified at the expense of the red side (E < 14).t For a general, nonlinear potential one should use Equations (6.3) and (6.4) instead of (6.6) for an accurate calculation of the spectrum. The reflection principle is well known in spectroscopy (Herzberg 1950 ch.VII Tellinghuisen 1987) the review article of Tellinghuisen (1985) provides a comprehensive list of references. For a semiclassical analysis of bound-free transition matrix elements see Child (1980, 1991 ch.5), for example. 

Derived from linear approximation of the equations (3.37), the equilibrium correlation function (4.29), defines two conformation relaxation times r+ and r for every mode. The largest relaxation times have appeared to be unrealistically large for strongly entangled systems, which is connected with absence of effect of local anisotropy of mobility. To improve the situation, one can use the complete set of equations (3.37) with local anisotropy of mobility. It is convenient, first, to obtain asymptotic (for the systems of long macromolecules) estimates of relaxation times, using the reptation-tube model. 

Fig. 3.13. Phonon-mode frequencies of wurtzite-structure PLD-grown Mg Zni- O thin films with Ai-symmetry (panel a, triangles) and Fi-symmetry (panel b, triangles), and of rocksalt-structure PLD-grown Mg Zni- O thin films (circles in both panels) vs. x [43,62,72,74], Open and solid symbols represent TO- and LO-modes, respectively. The dashed lines are linear approximations of the rocksalt-structure phonon modes from [74], the solid lines represent MREI calculations for the wurtzite-structure phonon modes redrawn from [132]. The shaded area, marks the composition range, where the phase transition occurs. Reprinted with permission from [74]...

Let us examine now the effect of the excluded volume at low surface potentials. In the linear approximation of the Poisson—Boltzmann expression, the increase in the number of counterions in the vicinity of the interface equals the decrease in the number of co-ions. If the co-ions have a larger size, one expects the available volume near the surface to be larger than that in the bulk. As a result, a concentration of ions in excess to that predicted by the Poisson—Boltzmann equation is expected to occur in the vicinity of the surface, when the volume exclusion is taken into account. 

IV.3. Modification of the Double Layer Repulsion Due to the Finite Volume of Ions. In the linear approximation of the Poisson—Boltzmann equation, the potential between two surfaces, separated by the distance l is given by... 

Although the nonlinear system of equations can be solved numerically, here we focus on the linear approximation of the Poisson—Boltzmann equation (which is accurate for small values of the potentials ip, qip/kT 1). Because the average polarization of water is P(x) = m(x)/v,h in this approximation eqs 1 and 2 become... 

A. Linear Approximation. The linear approximation of the Poisson—Boltzmann equation (s,mh.(ey> IkT) == eipl kT) is in general accurate only for small potentials (ip < kTle). However, it offers insight into the qualitative behavior, even at larger potentials. In the linear approximation, the Poisson—Boltzmann equations (4) become... 

This implies that even at moderate electrolyte concentrations (ce = 0.01 M, 2d = 30 A) and small dissociation fractions rj, ip(0) > kTle and the linear approximation of the Poisson—Boltzmann equations fails. An approximation, which is more accurate at large potentials, is suggested below. 

For small charges of the brush or large ionic strengths, one can employ the linear approximation of the Poisson— Boltzmann equations (eqs 12) with the boundary conditions (eqs 32). The solution that obeys the continuity... 

As derived from Table 5.2, the ratio r = C (298)/C (298) shows a standard deviation of 7% from the average value r = 1.32. This ratio will decrease, however, with increasing temperature, as the slope of C is steeper than that of Cj,. The linear approximations of the curves for C and Cp as a function of temperature (Eqs. (5.7) and (5.8)) may be used for estimating C and Cp at the melting point. The ratio Cj,/C at the melting point shows a standard deviation of 7% from the average value r = 1.10. This can also be seen from Table 5.2. 

Figure 86 Hypothetical trap potentials localized at the electrode in the absence (a) and in the presence (b) of image and external electric fields. Dashed lines represent a linear approximation of the trap potential in (a), and the potentials in the absence of trap in (b). Plots of logj vs. F1/2 according to Eq. (213) (c) parametric in at 4 x 106 cm-1 (1), 2 x 106 cm-1 (2), K cnT1 (3), 5 x K cnT1 (4) and at = 0 (5). After Ref. 361. Copyright 1989 Jpn. JAP, with permission.

The linear approximation of the Boltzmann distribution that allows one to calculate the time averaged density of opposite charge surrounding the central ion, is assumed to hold, and this is actually true only for very low ionic concentrations. 

Linear and non-linear approximations of the higher orders Therefore, condition (9.128) holds if... 

Quasi-linear approximation of the modified Green s operator We will obtain a more accurate approximation even on the first step if we assume that the anomalous field E inside the inhomogeneous domain is not equal to zero, as it was supposed in the previous section, but is linearly related to the background field E by some tensor A ... 

We call this approximation a quasi linear approximation of the first order Eq or a modified quasi-linear approximation (MQL) E)(fQ for an anomalous field ... 

Expression (9.79) for the quasi-linear approximation of the anomalous field can be written in discrete form as... 

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