Curvature quadratic function

We postulate that the free energy G of a liquid crystal specimen in a particular configuration, relative to its energy in the state of uniform orientation, is expressible as the volume integral of a free-energy density g which is a quadratic function of the six differential coefficients which measure the curvature ... 

Another way to interpret the procedure described is that the curvature along an arbitrary direction, in the surface a = r -f- wr, is a quadratic function of the values of i and m. Diagonalizing this quadratic form, subject to the constraint that a is a unit vector, is mathematically equivalent to the minimization/maximization of k. Thus, the extremal curvatures, Ka and Kb are determined by the extremal values of I and w, which we denote as I and m. For directions on the surface close to these extremal values, the expansion of the curvature as a function of ( — ) and (m — m ) has no linear terms since... 

The correction of the ln(k) to %B curves can also be described with a hyperboUc function [13], which has the advantage that it can be simply integrated. This is taken into account by a curvature factor a. The quadratic function proposed by Schoenmakers [12] sometimes shows a (physically implausible) upturn of ln(k) in the upper % -region, which however usually lies in the region where the substances have already been eluted. 

When the curvature is appreciable it may be necessary to use curve-fitting to estimate the initial slope accurately. Some methods based on integrated rate equations have been described, but for most purposes it is sufficient to use a generic equation such as a quadratic function to describe the progress curve. If this is done there u e several pitfalls to be avoidoL... 

One optical feature of helicoidal structures is the ability to rotate the plane of incident polarized light. Since most of the characteristic optical properties of chiral liquid crystals result from the helicoidal structure, it is necessary to understand the origin of the chiral interactions responsible for the twisted structures. The continuum theory of liquid crystals is based on the Frank-Oseen approach to curvature elasticity in anisotropic fluids. It is assumed that the free energy is a quadratic function of curvature elastic strain, and for positive elastic constants the equilibrium state in the absence of surface or external forces is one of zero deformation with a uniform, parallel director. If a term linear in the twist strain is permitted, then spontaneously twisted structures can result, characterized by a pitch p, or wave-vector q=27tp i, where i is the axis of the helicoidal structure. For the simplest case of a nematic, the twist elastic free energy density can be written as ... 

In Figure 6.3.2 note the large drop in pressure from the reservoir to the die. Laun (1983) has shown that this entrance pressure drop is in good agreement with that measured by capillary rheom-etry. Notice also the curvature in the pressure versus distance data. We saw curvature with different L/R capillaries in Figure 6.2.9. It is caused primarily by the pressure dependence of viscosity. We can analyze for pressure dependence by fitting the profiles with a quadratic function... 

We have already seen that modeling of the dissociative ET theory, in its more simple form, leads to a quadratic equation relating AG to AG° through equation (7). The kinetic sensitivity of a simple process on AG° is described by the transfer coefficient a, which is a linear function of AG° (equation 52). An important feature of equation (52) is that a is expected to be 0.5 for AG° = 0, less than 0.5 for favored ETs and larger than 0.5 for unfavored ETs. The second derivative (equation 53) describes the curvature of the parabola (or the slope of the a—AG° plot). 

The quadratic model is an improvement on the linear model since it gives information about the curvature of the function and contains a stationary point. However, the model is still unbounded and it is a good approximation to fix) only in some region around xc. The region where we can trust the model to represent fix) adequately is called the trust region. Usually it is impossible to specify this region in detail and for convenience we assume that it has the shape of a hypersphere s <, h where h is the trust... 

Can display local quadratic convergence Requires construction and factorization of preconditioner Performance may be slow for highly nonlinear functions when directions of negative curvature are detected repeatedly... 

Examination of the fit of the straight line to the data points (as characterized by the residuals) shows that the points tend to be a little below the line near the ends and a little above near the middle, as if the correct fitting function should have a small amount of curvature. Let us now consider a quadratic model, which may be justified by a more refined theory or may be purely empirical. Accordingly a new fit was made with the function... 

Simpson s rule approximates the curvature of the function by means of a quadratic equation. To evaluate the coefficients of the quadratic requires the use of the y values for three adjacent data points. The x values must be equally spaced. 

The quadratic model (Eq.3.3) allowed the generation of the 3-D response surface image (Fig. 3.5) for the main interaction between injection time and voltage. The quadratic terms in this equation models the curvature in the true response function. The shape and orientation of the curvature results from the eigenvalue decomposition of the matrix of second-order parameter estimates. After the parameters are estimated, critical values for the factors in the estimated surface can be found. For this study, a post hoc review of our model... 

The columns X, Xi and X2 can be seen to be identical for the factorial experiments. Following the same reasoning as in chapter 3, we conclude that the estimator bo for the constant term, obtained from the factorial points, is biased by any quadratic effects that exist. On the other hand the estimate b o obtained only from the centre point experiments is unbiased. Whatever the polynomial model, the values at the centre of the domain are direct measurements of Pq. So the difference between the estimates, bo - b o, (which we can write as U+22 using the same notation as in chaptw 3) is a measure of the curvature P, + P22- The standard deviation o o is s/v 8 (as it is the mean of 8 data of the factorial design) and that of b o is s/ /2. (being the mean of 2 centre points). We define a function t as ... 

The Hardin Jones plot for some presumably homogeneous cohorts of cancer patients shows some curvature, such as to suggest a moderate amount of heterogeneity. A reasonable assumption is that there is an error function distribution of the activation energy of the rate constant, a, about a mean value of the activation energy corresponding to an intermediate value a<, of the rate constant. This assumption leads on expansion and integration to the introduction of a quadratic term ... 

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