Parametric distortion

The problem with the current interpretation is that the symmetric orthogonalization is explicitly or implicitly assumed in the calculation of integrals, but the subsequent parametrization distorts this basis. If one subjects the OAOs obtained to a back transformation ... 

To describe the X-ray imaging system the projection of 3D object points onto the 2D image plane, and nonlinear distortions inherent in the image detector system have to, be modelled. A parametric camera model based on a simple pinhole model to describe the projection in combination with a polynomal model of the nonlinear distortions is used to describe the X-ray imaging system. The parameters of the model are estimated using a two step approach. First the distortion parameters for fixed source and detector positions are calculated without any knowledge of the projection parameters. In a second step, the projection parameters are calculated for each image taken with the same source and detector positions but with different sample positions. 

The exact nature of the axial field is irrelevant within this model — for example, it could be any of the distortions illustrated in Fig. 3 — and its effect is represented parametrically by a splitting between the orbital doublet and orbit singlet arising from the triplet 2 T2 term. The splitting, A in Fig. 4 (defined conventionally as positive if the singlet lies lowest), is thus defined, not in terms of the operator... 

The idea of the phase plane3 is to let the time be a parametric variable along the curve (u(t), v(r)) in the u, v plane. The equations are often autonomous (i.e., the right-hand sides are not functions of t) but, when they are not, they can be made so by adding w(t) = t. Then the third differential equation is w = 1. If necessary, a nonvanishing function of F and G, for example K(u, v), can be divided into each equation. This distorts the time that must be recovered from a third equation. A particularly useful transformation of this sort is given by K2 = F2 + G2, for then the independent variable is the path length, s, and... 

Quantitative studies by means of parametric statistical methods are, however, often very unreliable because of high environment-related variations very often amounting to several orders of magnitude [FORSTNER and WITTMANN, 1983 EINAX, 1990], In other words environmental data sets often contain values which are extremely high or low, i.e. they are outliers in the statistical sense. Also, because environmental data are often not normally distributed, the application of parametric statistical methods results in distorted reflections of reality. 

Up to now we have seen how lattice distortions are detected and characterized. This does not provide a direct observation of the molecular translations, rotations, and deformations associated with the distortion. However, for a few compounds it has been possible to measure a large enough number of satellite or superstructure reflections so that the distorted structure can be parametrized and refined (rigid-body or full structural study). We consider below four examples, taken from materials selected in Section IV. A, which show that such studies are not easy and that the data collection requires special attention. Indeed, it is generally difficult to measure enough satellite reflections, especially if several kinds of the latter coexist (e.g., 2kp and 4kF satellites, high-order satellites, etc.). 

One of the difficulties of quantitative imaging is that each of the steps that result in parametric mapping is closely dependent on the context, and especially on the model and the distortions induced by the measurement protocol. To our knowledge, no software package is currently available that will manage all the tasks described in this chapter. As efficient softwares for such optimization and reconstruction becomes more generic and commonly available, such techniques will be certainly applied with increasing frequency and efficiency. 

Spectroscopists often write Dj 3 Dg, risking confusion with the potential-well depth Dg in Fig. 14.1.) Centrifugal distortion is usually not very significant until rotational quantum numbers around J = 30 are reached. A more important effect is the dependence of the rotational constant on the vibrational quantum number. This can be parametrized as... 

Elementary substitution reactions of type I + R2R3 -> R1R2 + R3. with Rk a molecular group, have been described in the context of the coupled-channel method by Brodsky and Levich (1973). These authors introduced distortion potentials for reactants and products and a parametrized, isotropic potential coupling. In practice, transition amplitudes were calculated... 

Attempts to represent the three-body interactions for water in terms of an analytic function fitted to ab initio results date back to the work of dementi and Corongiu [191] and Niesar et al. [67]. These authors used about 200 three-body energies computed at the Hartree-Fock level and fitted them to parametrize a simple polarization model in which induced dipoles were generated on each molecule by the electrostatic field of other molecules. Thus, the induction effects were distorted in order to describe the exchange effects. The three-body potentials obtained in this way and their many-body polarization extensions have been used in simulations of liquid water. We know now that the two-body potentials used in that work were insufficiently accurate for a meaningful evaluation of the role of three-body effects. 

Aim and Background. The idea of the chapter is to study topological characteristics of protein water shell, to compare fibrillar and globular protein water shell structures and to discover if some topological invariants exist in H-bond water network of protein water shell. We consider H-bond water network from the graph theory point of view [1, 2], In addition we compare the obtained stmctures with ideal parametric bonded-water of Buhenkov s model [3], to understand if ideal or distorted th-cycles exist in water shells. 

We use the Bulienkov parametric bound water model to describe the protein water shell. Water molecules are bonded into H-bond network. This H-bond network can be performed as a system of hexacycles in twist-boat conformation. The twist-boat hexacycles provide non-Euclidean geometry parameters as it should be in crystal stracture such an ice [18]. In ice stracture the internal parameters of all hexacycles are equal-intermolecular distances, valence and torsion angles are constant and can vary only by a thermal motion. So if any hexacycle system is constructed using only twist-boat pattern then geometrical parameters must distort [23]. 

Another factor hampering application of MM to biological molecules is that the metal center is often found in an unusual coordination environment and there are insufficient related model complexes upon which to build a reasonable MM parametrization. For example, the type I blue copper center contains a highly distorted CUN2S2 center which is neither planar nor tetrahedral. This places extra demands on the MM approaeh. However, the development of sophistieated DFT methods are enabling these unusual struetures to be modeled from first principles. It should then be possible to develop MM parameters based on the DFT data. ... 

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