Analysis tensorial

The other source of an effective electric field dependence of the diffusion coefficient is due to hydrodynamic repulsion. As the ions approach (or recede from) one another, the intervening solvent has to be squeezed out of (or flow into) the intervening space. The faster the ions move, the more rapidly does the solvent have to move. A Coulomb interaction will markedly increase the rate of approach of ions of opposite charge and so the hydrodynamic repulsion is correspondingly larger. It is necessary to include such an effect in an analysis of escape probabilities. Again, the force is directed parallel to the electric field and so the hydro-dynamic repulsion is also directed parallel to the electric field. Perpendicular to the electric field, there is no hydrodynamic repulsion. Hence, like the complication of the electric field-dependent drift mobility, hydro-dynamic repulsion leads to a tensorial diffusion coefficient, D, which is similarly diagonal, with components... 

In this Appendix, the equivalence of the diffusion equation treatment and the molecular pair analysis is proved (see Chap, 8, Sect. 3.2) for the situation where there is a potential energy E/(r) between the reactants and the diffusion coefficient is tensorial and position-dependent. This Appendix is effectively a generalisation of the analysis of Berg [278]. The diffusion equation has a Green s function G(r, f r0, t0) which satisfies eqn. (161)... 

In many cases, the spectra recorded for the condensed phase are similar to those recorded in the liquid phase, but they usually contain a wider range of information than is available in liquid NMR spectroscopy. The solid state represents the best environment for the investigation of intermolecular interactions. Analysis of the tensorial nature of the chemical shifts provides subtle structural information. Strategies based on dipolar recoupling and /-coupling indicate a number of ways in which direct and indirect coupling constants can be measured, to yield direct structural constraints. This approach, combined with advanced theoretical calculations, traces new trends in structural studies of the condensed matter. 

Gregorio Ricci-Curbastro. Tensorial Analysis and absolute differential calculus. 

A nonspherical particle is generally anisotropic with respect to its hydro-dynamic resistance that is, its resistance depends upon its orientation relative to its direction of motion through the fluid. A complete investigation of particle resistance would therefore seem to require experimental data or theoretical analysis for each of the infinitely many relative orientations possible. It turns out, however, at least at small Reynolds numbers, that particle resistance has a tensorial character and, hence, that the resistance of a solid particle of any shape can be represented for all orientations by a few tensors. And the components of these tensors can be determined from either theoretical or experimental knowledge of the resistance of the particle for a finite number of relative orientations. The tensors themselves are intrinsic geometric properties of the particle alone, depending only on its size and shape. These observations and various generalizations thereof furnish most, but not all, of the subject matter of this section. 

Gabriel Kron, Tensorial analysis of integrated transmission systems. AIEE Transactions, Vol. 70, pages 1239 to 1248, 1951. 

In his detailed analysis of Dirac s theory [6], de Broglie pointed out that, in spite of his equation being Lorentz invariant and its four-component wave function providing tensorial forms for all physical properties in space-time, it does not have space and time playing full symmetrical roles, in part because the condition of hermiticity for quantum operators is defined in the space domain while time appears only as a parameter. In addition, space-time relativistic symmetry requires that Heisenberg s uncertainty relations. 

Shear Modulus. The shear modulus determined from torsion measurements exhibits some dependence on temperature. For the Kevlar composite, the increase was 1.5, and for carbon fiber composite it was 1.2, from 293 to 4.2 K. Of course, both the damping and storage shear moduli represent tensorial quantities, and this must be included in the analysis. For anisotropic fibers, both the tensorial quantities of the fibers and those of the composite are involved. Here, only one tensor element, which was expected to be sensitive to temperature, was considered. 

It is important to stress once more that pSR measures the fluctuations of the localflelds at the muon site and not directly the fluctuations of the spins creating those fields. This means in essence that for a quantitative analysis (for TF as well as for ZF measurements) the exact relations between the individual components of lanthanide spin correlations S t)SfQ)) and the muon field correlations B t)B jS)) must be worked out for each particular muon spin -> lanthanide crystal geometry. This step is often overlooked. The general tensorial relation between spin and field correlation has been given by Dalmas de Reotier et al. (1996). 

Flexoelectricity is a basic mechano-electric phenomenon in liquid crystal physics. The first hints for the existence of a curvature-polarization tensor in liquid crystals, although with different sjmunetry transformation properties, can be foimd in a manuscript by Freedericksz (1940) that made use of the tensorial analysis method of general relativity and was published only... 

Within the f shell, the magnetic interactions and lend themselves to the same kind of analysis that Racah (1949) carried out for the Coulomb interaction (see section 4.3). Just as the latter can be described in tensorial terms by where the... 

Andraud et have made a theoretical analysis, using a CNDO/S method, of polyenic molecules of octupolar Csh symmetry. The results have been analysed in a tensorial formalism to elucidate the optimum arrangement of groups for high in octupolar molecules. 

Santra, R. and Greene, C.H., Tensorial analysis of the long-range interaction between metastable aUcaline-earth-metal atoms, Phys. Rev. A, 67, 062713, 2003. 

A similar DIPOLE-based procedure could evidently he extended to higher-order hyperpolarizability components. However, the number of tensorial components rises steeply with tensor order, and the numerical differencing problems associated with accurate evaluation of limiting ratios such as (6.24) become increasingly challenging. Practical DIPOLE-based analysis of such higher-order polarizability properties may therefore be limited to the leading few components. 

The two-particle nature of Coulomb interaction in equation (10.27) is the reason that among the third-order contributions to the transition amplitude, in addition to one particle effective operators (as in the standard J-O approach), two particle objects are also present. However, the numerical analysis based on ab initio calculations performed for all lanthanide ions, applying the radial integrals evaluated for complete radial basis sets (due to perturbed function approach), demonstrated that the contributions due to two-particle effective operators are relatively negligible [11,44-58]. This is why here they are not presented in an explicit tensorial form (see for example Chapter 17 in [13]). At the same time it should be pointed out that two-particle effective operators, as the only non-vanishing terms, play an important role in determining the amplitude of transitions that are forbidden by the selection rules of second- and the third-order approaches. This is the only possibility, at least within the non-relativistic model, to describe the so-called special transitions like, 0 <—> 0 in Eu +, for example, as discussed above. 

It is possible to identify all of these different contributions to the intensity parameters when deriving their tensorial form explicitly at the second, third, or even higher orders of perturbation approach. It is impossible however to go backwards and select parts of fitted parameters and assign to them a particular physical meaning or interpretation. In general, the parameters Tlx represent the overall picture of possibly important mechanisms that affect the/ <—> f transition amplitudes, without any specification about the order of perturbation, since this particular formalism does not play any role at the point of numerical analysis of their fitted values. 

At the light of recent studies about the origin shift which can be performed on the QOS tag sets [24], several interesting results have been obtained. At the same time, some work has been done, using this origin shift point of view, on stereoisomers [75]. These previous studies are relevant at this stage of the present study and thus some analysis will be performed on the question of binary QOS and the coimection with the TD tensorial representation of such systems when origin shifts are done on the constituent tag set DF. 

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