Banana shaped model

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. 

Figure 3.5 Example sum of squares contour plots, (a) contour plot of a linear model illustrating that the contours for such a model are perfecdy elliptical with the global minimum at the center, (b) example of a well defined nonlinear model having a single global minimum and nearly elliptical contours, (c) contour plot of a nonlinear model showing banana -shaped contours indicative of ill conditioning, (d) contour plot of a nonlinear model with multiple solutions. All nonlinear models were parameterized in terms of a and (3. Upper right and bottom left examples were suggested by Seber and Wild (1989). The symbol ( ) indicates global minimum. M indicates local minimum.

T. Imase, S. Kawauchi, J. Watanabe, Conformational analysis of 1,3-benzenediol dibenzoate as a model of banana-shaped molecules forming chiral smectic phases,... 

FIGURE 4.33. Calculation of the flexoelectric coefficients for (a) pear-shaped molecules (b) banana-shaped molecules and (c) the quadrupole model of the flexoelectric effect [200]. 

The thermotropic liquid crystal, 4,4 -diheptylazoxybenzene (HAS), exhibiting isotropic, nematic and smectic phases, has been studied through e NMR. The temperature dependence of e chemical shifts and spin-lattice relaxation times of the Xe gas dissolved in HAS showed clear signatures of the phase transitions. Theoretical models have been used to understand the influence of the different phases on the isotropic and anisotropic parts of the chemical shielding. From the studies it is also inferred that in the smectic phase, Xe atoms preferentially occupy interlayer spacings rather than the interiors. Bent-core or banana-shaped molecules display an array of novel chiral liquid crystalline phases. NMR studies on two of the banana core moieties have been analyzed using ab initio structure calculations and the steric inertial frame model. ... 

Integration of the relevant signals revealed that this new assembly contains six units of propandiurea—an unprecedented nnmber of spacer units since all other extensions involve multiples of four units. Detailed NMR studies and molecular modeling revealed an unusual banana-shaped structure for this new assembly. The unusual shape suggested the potential of the structure for shape-selective encapsulation of rigid guests and this proved to be the case. Structures with kinks at the middle... 

Molecular models for (a) pear-shape molecules (b) banana-shape molecules, used for calculation of flexoelectric coefficients. 

This is the exact confidence region with approximate confidence levels. The confidence region considers the effect of varying all parameters. For linear models, the confidence regions assume an elliptical shape, whereas the region is often more banana shaped for non-linear models. 

Nowadays, rods and discs are considered as model structures because some other unconventional shapes have been recently described such as bananas (bent-core), cones and rings. 

This phenomenon has been discovered in the liquid crystal phases consisting of so-called banana (or bent-core) shape molecules [17, 27]. A mechanical model in Fig. 4.39a illustrates the idea. Each of the two dumb-bells has symmetry Do h with infinite number of mirror planes containing the longitudinal rotation axis and one mirror plane perpendicular to that axis. Imagine now that one of the dumb-bells is lying on the table and we try to put another one on the top of the first one parallel to... 

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